*NAVAL POSTGRADUATE SCHOOL · within the settings of fractals and chaotic dynamical systems. This...
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*NAVAL POSTGRADUATE SCHOOL JI
Monterey, California
AD-A246 502
THESISFRACTALS AND CHAOS
by
Philip Frederick Beaver
June 1991
Thesis Advisor: Maurice D. WeirCo-Advisor: Ismor Fischer
Approved for public release; distribution is unlimited.
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ATASANDAhow_ m Nom m I P No Task No IWo*i Umk Amci==, No11 Title (Include Security Classification) rACT AND CH OS12 Personal Author(s) Beaver, Philip F.13a Type of Report 13b Time Covered 14 Dame of Report (year, noinhday) 15 Page CountMaster's Thesis From To June 1991 17916 Suppleme y Notation The views expressed in this thesi are those of the author and do not reflect the officialpolicy or position of the Department of Defense or the U.S. Government.17 Cosati Codes 18 Subject Terms (coninue on reverse sfnecessary and identif by block number)Field Group subgroup Fractal geometry and chaotic dynamical systems
19 Abstract (continue on revern if necesuary and identiy by block numberThe study of fractal geometry and chaotic dynamical systems has received considerable attention in the past
decade. Motivated by the interesting computer graphics produced by these fields, mathematicians have attemptedto formalize the theoretical structure of the results, physicists have attempted to apply the theory to real worldphenomena, and laymen have enjoyed much of the popular literature and television programs the field hasfostered. Unfortunately, the mathematics associated with these subjects has made them inaccessible to mostundergraduates, even if they have a strong background in mathematics. This thesis presents the basic ideas offractal geometry and chaotic dynamical systems in a setting that can be understood by undergraduate students whohave had a course in advanced calculus. We hope it will allow them to gain an appreciation of the fields andmotivate them to pursue further study.
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Fractals and Chaos
by
Philip Frederick BeaverCaptain, United States Army
B.S., United States Military Academy, 1983
Submitted in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE IN APPLIED MATHEMATICS
from the
NAVAL POSTGRADUATE SCHOOLJune 1991
Author: Philip Frederick Beaver
Approved by: Ali A /ffS Maurice D. Weir, T7hesis Advisor
Ismor Fischer, Thesis Co-Advisor
Harold M. Fredricksen, Chairman
Department of Mathematics
ii
ABSTRACT
The study of fractal geometry and chaotic dynamical
systems has received considerable attention in the past
decade. Motivated by the interesting computer graphics
produced by these fields, mathematicians have attempted to
formalize the theoretical structure of the results,
physicists have attempted to apply the theory to real world
phenomena, and laymen have enjoyed much of the popular
literature and television programs that the field has
fostered. Unfortunately, the mathematics associated with
these subjects has made them inaccessible to most
undergraduates, even if they have a strong background in
mathematics. This thesis presents the basic ideas of fractal
geometry and chaotic dynamical systems in a setting that can
be understood by undergraduate students who have had a course
in advanced calculus. We hope it will allow them to gain an
appreciation of the fields and motivate them to pursue
further study.
Aocessicn For
D' Ly'r 0d I
W. -o
[iii 1 r'LL•,
TABLE OF CONTENTS
KI BASIC CO CPS.-----..A. INTRODUCTION....................................................4..
A . INTRODUCTION ...................................................................................... ...... 4
B. METRIC SPACES ...................................................................................... 4
C. ITERATED FUNCTION SYSTEMS .............................................................
D. CODE SPACE ............................................................................................. 13
E. THE CANTOR SET ................................................................................. 15
HI. FRACTALS ------------------------------------------------------------ 19
A. THE SETTING FOR FRACTALS ........................................................... 19
B. CONTRACTION MAPPINGS ............................................................. 22
C. AFFINE TRANSFORMATIONS OF THE PLANE ........................... 25
D. CONTRACTION MAPPINGS OF THE SPACE gi(X) ....................... 29
E. CREATING FRACTALS THROUGH ITERATED FUNCTIONSYSTEMS ........................................................................................................ 33
1. The Cantor Set ................................................................................. 33
2. Fractals in Two Dimensions ........................................................... 36
3. Condensation Sets .......................................................................... 42
4. A Fractal Tree .................................................................................. 43
F. APPLICATIONS OF FRACTALS TO COMPUTER GRAPHICS .......... 46
G. THE ADDRESSES OF POINTS ON FRACTALS ................................. 50
H. FRACTAL DIMENSION ....................................................................... 60
I. EXPERIMENTAL DETERMINATION OF FRACTALDIM ENSION .............................................................................................. 69
J. THE KOCH SNOWFLAKE .................................................................... 73
K. APPLICATIONS OF FRACTAL GEOMETRY ..................................... 75
iv
A. INTRODUCTION .........................................................................................77
B. GRAPHICAL ANALYSIS OF FIXED POINTS OF MAPS ................. 79
C. MAPS OF THE CIRCLE .......................................................................... 82
D. CHAOTIC DYNAMICAL SYSTEMS ................................................... 85
E. TOPOLOGICAL CONJUGACY .............................................................. 89
F. CHAOTIC DYNAMICS ON CODE SPACE .......................................... 90
G. NEWTON'S METHOD FOR X2 = -1 ................................................... 92
H. THE QUADRATIC FAMILY OF MAPS .................................................... 96
L BIFURCATIONS .......................................................................................... 108
J. SARKOVSKIrS THEOREM ...................................................................... 118
K. THE QUADRATIC FAMILY REVISITED ............................................... 123
L JU LIA SETS ................................................................................................... 131
M. THE MANDELBROT SET ......................................................................... 137
N. THE SMALE HORSESHOE ........................................................................ 144
0. THE HENON MAP ..................................................................................... 149
P. THE LORENZ EQUATIONS ..................................................................... 155
LIST OF REFERENCES.. . . .. .. . . . .168
INITIAL DISTRIBUTION UST --. ..... . .. 171
ACKNOWLEDGMENT
I would like to express my appreciation to a number of individuals whohave played a role in the development of this thesis. I am particularlygrateful to the following:
Professor Maurice D. Weir, who first suggested the subject of this thesis,and who developed the curriculum that provided the necessary backgroundfor me to write it. He continued to guide me throughout the preparation ofthe thesis, and spent countless hours encouraging, proofreading, andmentoring.
Professor Ismor Fischer, who led me through a course on chaoticdynamical systems, provided technical guidance and expertise during thepreparation of this thesis, and who remained patient throughout the entireprocess.
Professor Aaron Schusteff, who started me on these subjects by teaching acourse on fractal geometry and topology, and who remained an essentialsource of technical information during my further study of chaotic dynamicalsystems and throughout the preparation of this thesis.
Finally, Mr. David Beaver of The Automation Group, who introducedme to the idea of chaos by sending me James Gleick's book on the subject, andwho volunteered numerous hours of word processing advice, without whichI could not have prepared this document.
As usual, however, I retain sole responsibility for all the errors, mistakes,and other deviations from the truth that may be contained within this thesis.
Philip BeaverMonterey, CaliforniaJune 1991
vi
I. INTRODUCTION
The subjects of fractals and chaos have attracted considerable attention in
the last decade. This interest ranges from a "cult following" of laymen who
are intrigued by the intricate computer graphics associated with the fields, to a
rigorous mathematical treatment of the subjects by topologists and experts in
dynamical systems, and to applications of these results to the real world by
engineers and physicists. However, these subjects have been almost wholly
inaccessible to undergraduates because of the level of mathematics required to
study them. This thesis presents the subjects of fractals and chaos in a setting
that can be understood by a typical undergraduate student with a solid
background in mathematics through advanced calculus.
The subjects of fractais and chaos are not new. The German
mathematician Georg Cantor (1845-1918) knew about fractals, and the French
mathematician Jules Henri Poincare (1854-1912) knew about chaos in
dynamical systems in the late nineteenth century. Additionally, the French
mathematicians Pierre Fatou (1878-1929) and Gaston Julia (1893-1978) knew
about Julia sets in the 1920s. However, it was not until the 1970s that high-
speed computers allowed others to see what these men had discovered and to
recognize the true potential of these fields. The growth of these fields has
resulted in significant scientific advances in the past decade. While the
discovery of quantum mechanics and relativity had a profound impact on
very specialized areas of science, fractals and chaos have had a universal effect
on the whole scientific community. Until recently, science had become so
1
specialized that, not only were physicists and mathematicians not
communicating with each other, but molecular biologists were not
communicating with population biologists either. The science of chaos has
served to bring together the entire scientific community, including physicists,
electrical engineers, mechanical engineers, biologists, economists,
astronomers, meteorologists, medical scientists, and of course,
mathematicians.
To explain briefly, fractals and chaos help to describe the universe.
Natural objects, from crystals, plants and geological formations to weather
patterns and galaxies, seem to have a fractal-like structure which eludes
description by traditional geometric means. Additionally, dynamical
(changing) systems in the real world, from turbulence in fluids to fluctuating
economic trends and unpredictable weather patterns that previously defied
description, are now being understood through the mathematics of chaos.
While these fields are still in their infancy, the potential they have already
demonstrated seems very promising in increasing our understanding of the
physical world.
The mathematics required to understand this thesis includes basic courses
in calculus and linear algebra. Additionally, due to some of the examples, a
familiarity with numerical analysis is helpful, but not required. Most results
(specifically in the fractals portion) are presented in the Euclidean plane, with
a brief mention of more abstract spaces where applicable. Mathematical
proofs that can be understood with this basic background are presented in
their entirety. Results requiring more advanced mathematics are referenced.
Where applicable, results are explained and interpreted in a basic setting. The
2
sections on Julia sets and the Mandelbrot set require a small amount of
complex variable theory. Finally, only discrete dynamical systems are treated
(with the exception of the section on the Lorenz equations), so a knowledge of
differential equations is not required.
None of the material in this thesis constitutes original research. Instead,
it is a synthesis of some of the best written works on the subjects of fractals
and chaos, interpreted and presented in such a way as to be understood by a
typical undergraduate with a basic mathematical background. Hopefully, it
will serve to stimulate further interest and deeper study in these exciting new
fields.
Specific references are listed in the sections to which they apply, and all of
the figures and examples taken directly from references are so cited as they
appear. However, we note that the two primary references for this thesis are
Barnsley (1988) for the chapter on fractals, and Devaney (1989) for the chapter
on chaos. Additionally, the references used for general information
throughout the thesis are Ross (1980) for advanced calculus, Anton (1987) for
linear algebra, Churchill (1990) and Boas (1987) for complex variables, and
Giordano/Weir (1991) and Arnold (1987) for differential equations.
3
II. BASIC CONCEPTS
A. INTRODUCTION
There are several recurring mathematical concepts that appear in the
study of both fractals and chaos. An understanding of these basic concepts
simplifies this analysis. Instead of presenting them as they arise, we next
discuss these concepts as background material. They are presented in a strict
mathematical setting which is easily adapted to their further treatment
within the settings of fractals and chaotic dynamical systems. This
background material is common to most of the references cited (either as
presented material or assumed knowledge), but the presentation here most
closely follows those of Barnsley (1988) and Devaney (1989).
B. METRIC SPACES
We first introduce the concept of metric spaces. Although the notion of a
metric space defined below may seem abstract and unfamiliar, the most
common examples of the real line and the Euclidean plane are encountered
in calculus. Many results from fractal geometry can be applied to general
metric spaces, but for the purpose of simplifying them, they are presented in
their most natural or familiar settings (such as the real line or the Euclidean
plane).
A space is simply a collection of objects (a set) with some additional
structure imposed on it. In a metric space, the additional structure is a metric,
or distance function, which relates every two elements x and y in the set to a
4
unique nonnegative real number d(x, y), called the distance between them.
Any distance function (or metric) must satisfy the following three axioms for
all x, y, z in the space:
1. Symmetry: d(x, y) = d(y, x);
2. Positive Definiteness: d(x, y) > 0, and d(x, y) = 0 c x = y;
3. Triangle Inequality:. d(x, y) < d(x, z) + d(z, y).
A metric space is denoted by (X, d), where X is the set and d is the
particular distance function. The metric space (R, dE) is the real line R with
d(x, y) = I x - y I. The space (R2, dE) is the Euclidean plane, where X = RXR
and d(x, y) = [(xl - yl)2 + (x2 - y2)2]1/ 2 . Whenever possible, we confine our
examples to the real line or the Euclidean plane with these metrics.
We also need the concept of closure, a thorough treatment of which can
be found in most advanced calculus texts. The closure of a set is the
intersection of all closed sets containing that set. An equivalent definition,
often presented as a theorem, is that if a sequence of points in a set S
converges to a point x in the space, then x is in the closure of S. Thus,
every set is a subset of its closure. As an example, consider the set Q of
rational numbers. The number d is not in Q, but the sequence of rational
numbers 1.4,1.41,1.414,1.4142,... converges to -F2, so F2 is in the closure of
Q (in fact, every real number is in the closure of Q).
A concept frequently associated with closure is that of denseness. A set S
is dense in a metric space (X, d) if for every point x e X and all c > 0, there
exists a point s e S such that d(s, x) < e. For example, the rational numbers
form a dense subset of the real line. Another way to state this is that between
5
any two points on the real line, there exists a rational point. If a set S is
dense in the space X, then the closure of S is X.
Much of our work on fractals and chaotic dynamical systems involves a
topological property of some metric spaces called completeness. A Cauchy
sequence is a sequence in which the terms get arbitrarily close to each other,
i.e., for any e > 0, there exists a number N such that d(xn, Xm) < e for all
m, n > N. A complete metric space is one where any Cauchy sequence of
points in the space converges to a point in the space. An intuitive example is
to again consider the space of rational numbers and the sequence 1, 1.4, 1.41,
1.414, 1.4142,..., which converges to the irrational V2. Clearly, terms in this
sequence get arbitrarily close to each other; however, the limit value (42) is
not in the space of rational numbers, so the sequence does not converge to a
point in the space. This is to say that there are "holes" in the space of rational
numbers, hence it fails to be a complete space. The real line, on the other
hand, is complete, since all Cauchy sequences of real numbers do converge to
a real number. The spaces (R, d) and (R2, dE), are examples of complete
metric spaces. A rigorous development of the concept of completeness and
formal proofs of the completeness of the spaces just mentioned can be found
in most advanced calculus texts.
C. ITERATED FUNCTION SYSTEMS
Fractals can be created, and chaotic dynamical systems understood,
through an analysis of iterated function systems. A simple one-dimensional
case of such a system is demonstrated by entering a starting value in a
calculator and repeatedly activating a single function key (for example, the x2
6
key). The result of executing this idea, with a starting value of 2 and the x2
key, is 2, 22, 24, 28, 216, .... In this example, the system diverges to infinity
and demonstrates a regular and predictable behavior. Another example of an
iterated function system on a calculator is to use the 'Fx key and start with
any positive number. Regardless of the initial value, this system will
converge to 1. Unfortunately, not all iterated function systems are quite so
simple. The study of chaotic dynamical systems attempts to explain irregular
behavior in more complicated iterated function systems.
A function, or map, is a rule which assigns to each element in a specified
domain a unique element in a codomain. The usual notation for this concept
is f: D-+R, where f is the rule, D is the domain, and R is the codomain. A
real-valued function of a real variable is specified by f: R--R, and a function
mapping the closed unit interval to itself is written f: [0, 1]-40, 11. A function
f: R--R of the form f(x) = ax where a is a constant, is said to be linear; if it is
of the form f(x) = ax + b with a and b constants, it is called affine.
A function is one-to-one if each element in the range is the image of a
unique element in the domain; that is fAx) = f(y) implies x = y. A function is
onto if every element in the codomain is the image of at least one element in
the domain. A continuous function f: R--R is stated as f e C, and a
continuously differentiable function is stated as f e C1. A function that is
n-times continuously differentiable is expressed by f e Cn, and its nth
derivative is written as f(n)(x). If a function is one-to-one, then its inverse fH
exists according to the rule f 1(x) = y where f(y) = x. The domain of f-1 is the
range of f. A function which is continuous, one-to-one, and onto for which
H is also continuous is called a homeomorphism. If f is a
7
-homeomorphism and f and f-1 both are differentiable, then f is called a
diffeomorphism.
As previously mentioned, an iterated function system results from
repeatedly applying a function to an initial point x. The sequence x, f(x),
f(t(x)), f(f(f(x))),... is also written as x, f(x), f2(x), fP(x),. .... This sequence of
iterates is called the forward orbit of x under f, and is denoted by O+(x).
From our example above, the forward orbit of 2 under f(x) = x2 is 2, 4, 16,
256,. .... If f is a homeomorphism, f-1 exists, and the backward orbit of x
under f is the sequence x, f-1(x), f-2(x),... ,denoted by O-(x). Since f(x) =x2
is not a homeomorphism, there is no backward orbit under f. However, if
we let fRx) = 2x, then the backward orbit of 2 under f is 2, 1, 1/2, 1/4, ... A
point x for which fAx) = x is called a fixed point of f; a point x for which
fn(x) = x is called a periodic point of period n. The smallest positive integer
n such that fn(x) = x is called the prime period of x. A point x is said to be
eventually periodic if x is not periodic, but there exists an integer m such
that fro(x) is a periodic point. For a function f where the derivative is
defined on the entire domain, a point x is a critical point if f(x) = 0, for
example f(x) = x2 at x = 0. It is a degenerate critical point if it is a critical point
and Nf(x) = 0, where the prime notation denotes differentiation of the
function.
EXAMPLE. The following simple example illustrates a situation when a fixed
point is guaranteed to exist for a given function. Let f: [0, 1]-4[0, 1] be such
that f e C. Then f has at least one fixed point on [0, 1], as we now prove. If
fRO) = 0 or f(1) = 1, then 0 or I is a fixed point and we are done. Otherwise,
8
f(O) > 0 and f(1) < 1, and define g(x) = f(x) - x. Since g is the difference of two
continuous functions, g is continuous. Also g(0) > 0 and g(1) < 0. Hence, by
the Intermediate Value Theorem from elementary calculus, there exists a
point a e [0, 1] such that g(a) = 0. Therefore, f(a) = a, completing the proof
(see Figure 2.1). This proof can be extended to any continuous function
mapping a dosed interval [a, b] to itself.
V y-x
a .... .......
0 a X
Figure 2.1 The fixed point of a graph f. [0,11-40, 11.
We will frequently use a geometric or graphical analysis to investigate
iterated function systems. For example, the orbit of two points, p and q,
under fAx) = x2 is shown graphically in Figure 2.2. Clearly, I and 0 are fixed
points of fAx) A x2. The orbits of points greater than 1 diverge to infinity, and
9
"the orbits of points in the interval [0, 1) converge to 0. This example
suggests the notion of fixed points (or periodic points) as being "attracting" or"repelling."
0 p lq X
Figure 2.2 The orbits of two points p and q under f(x) = x2.
A periodic point p of prime period n (to include fixed points with n = 1)
is called hyperbolic if I (f)(p) I * 1. If I (fn)(p) I < I, then p is called an
attractor, and if I (fnf)'(p) I>1, then p is called a repellor. As in the above
example with fWx) = x2, the orbit of points near an attractor tend towards that
point, while the orbits of points near a repellor tend away from that point. A
precise definition of "near" will be given in the chapter on chaos. With
fRx) = x2, the derivative is f'(x) = 2x, so the derivative evaluated at the fixed
points 0 and 1 gives f(0)=0 and f(1)= 2. Therefore, 0 is an attracting
fixed point and I is a repelling fixed point. Values in the interval [0, 1) tend
10
"towards 0 and away from 1, while values in the interval (1, -) all diverge.
Hence, the point 0 "attracts" iterates and 1 "repels" them.
A graphical analysis can be very useful when analyzing fixed, periodic, or
eventually periodic points. The graphs in Figures 2.3 through 2.6 show
functions with fixed points, periodic points, eventually fixed points, and
eventually periodic points. While geometric constructions cannot replace
rigorous mathematical proofs, they are useful to demonstrate and provide
insight into phenomena that occur in iterated function systems.
y
f (a) ... ..........
0 a X
Figure 2.3 A fixed point a of f(x): f(a) - a.
I1
y --x
0 Pl P2 x
Figure 2.4 Periodic points of f(x): f(pl) = p2 and f(p2) = pl.
y-X
f2(p)
0 i f 2() X
Figure 2.5 An eventually fixed point p of f(x): fWf2(p)) = f2(p).
12
yAL x
0 p f 2(p) X
Figure 2.6 An eventually periodic point p of f(x): f2(p) is periodic.
D. CODE SPACE
A useful concept for identifying different points on fractals (frequently
called addressing points on fractals) and analyzing chaotic dynamical systems
is that of code space. We define 7,2 as the set of all infinite sequences of
binary digits sIs2s3... where si e (0, 1). We next define a distance function
for all s, te 12 byd(s, t) I Isi - til/2 i.
ii
Recalling the properties of a metric, it is easy to verify that this mapping
does define a distance function on 12 since I si - ti I ! 1.. Since the distance
between any two points in 12 is dominated by the convergent geometric
series
i-i 2
13
a unique and finite distance exists between any two points in the code space
(that is, d(s, t) is well-defined).
The concept of code space can be extended to IN, which consists of the
infinite sequences s1, s2, S3,. • ,where si e (0, 1, 2,3,..., N-1). Here the
distance function becomes4Wd(s, 0 Is=-tj1(
i-i
It is an easy exercise to show that the properties of a metric space are satisfied.
We will frequently perform an operation on code space known as the
shift map. The shift map a: 12-+12 is given by o(s1s2s3.• .) = (s2s3s4. .. ). The
shift map simply drops the first term in the sequence. Since a(0s2s3S4. •) =
O(1S2S4.. .), the shift map fails to be one-to-one.
In an iterated function system of the shift map a, U1, a2, g3,... on
the only fixed points are 000... and 111... ; the eventually fixed points are of
the form sls2...nOO0... and sls2. • .1... , and the periodic (or eventually
periodic) points consist of sequences having a repeating (or eventually
repeating) block of Os and Is. However, there are infinitely many points
which are neither fixed nor periodic. These points are the sequences which
have no repeating nor eventually repeating blocks of Os and Is.
We will use code space extensively and develop it in greater detail when
discussing fractals and chaotic dynamical systems. For now, this introduction
provides a brief exposure to the basic ideas of this important concept.
14
E. THE CANTOR SET
The Cantor set, which has traditionally served as an important
pathological example in analysis, has secured a more distinct role in the
studies of chaos and fractals. While it is actually one of the simplest examples
of a one-dimensional fractal, we first present it in its traditional construction
to ensure familiarization with the set before analyzing it in its new guise.
To construct the classical Cantor, or middle-thirds set, begin with the unit
interval [0, 1] and remove its middle third open interval (1/3, 2/3). Of the
two remaining line segments, [0, 1/3] and [2/3, 1], remove the open middle
third of each of them. Of the four remaining line segments, again remove
their open middle thirds, and continue this iterative procedure ad infinitum.
The remaining set of points, which is a subset of the unit interval, is called
the Cantor middle-thirds (or ternary) set. It is sometimes referred to as
"Cantor dust" because of the scattered configuration of the remaining points.
A picture of the construction of the classical Cantor set is shown in Figure 2.7.
The Cantor set dearly contains an infinite number of points because, at
the very least, the endpoints of the line segments left after each iteration
remain in the set. Nevertheless, the total amount of length removed from
the unit interval is equal to one. To see this, consider the length removed
during each iteration. In the first step, a segment of length 1/3 is removed;
in the second step, two segments each of length 1/9 are removed; and in the
third step, four segments each of length 1/27 are removed. More generally,
in the kth step, 2k-I segments each of length 1/3k are removed. Summing
the total lengths removed yields
15
0 1/3 2/3 1
- -
Figure 2.7 Constructing the dassical Cantor set by removing the middlethird through infinite iteration. The Cantor set consists of the points thatremain.
L f 2"kl(/)" = (/)1T . (2J3) n= (W1)[I/(1 - 2/3)] = 1.nfIl n=O)
So the total length removed is the entire length of the unit interval.
Another way of approaching the Cantor set is through ternary expansion.
Consider expanding each point in the unit interval in its base three form
0.xlx2x3..., where xi e (0, 1, 2). The value of each point is then xj(1/3) +
x2(1/3 2) + x3(1/33) +. ... There is a minor technicality with this approach,
because many points in the unit interval have dual ternary representations.
For example, the point 1/3 can be expressed both as .1000... and as .0222...
In fact, the endpoints of every interval in our previous construction of the
Cantor set, except for 0 and 1, have such a dual representation. The way we
remedy this is to always use the representation containing the repeating 2.
Now, all of the points between 1/3 and 2/3 have xj = 1, and these are the
16
points removed in the first step of our former c-nstruction. Similarly, the
points removed in the second step have the form xj = 0 or 2, and X2 = 1.
Continuing with this analysis, we see that the points remaining in the Cantor
set are precisely those that have no ls appearing in their ternary expansion.
Thus the Cantor set consists of all points in the unit interval having
only Os or 2s in their ternary expansion. Moreover, every point which has
only Os or 2s in its ternary expansion is a member of the Cantor set. This
definition proves to be very useful when we analyze chaotic dynamical
systems. Note that the Cantor set contains no interval subsets because
between any two points, there is a point with a 1 in its ternary expansion. To
see this, assume that the Cantor set does contain an interval subset. We can
let the endpoints of this interval be XIX2.. .XnoXn+2... and X1X2.. .Xn2 Xn+2...,
where the xi agree for the first n digits, are different for the n+lst digit
(which means one must be a 0 and the other must be a 2), and after which
the digits can be arbitrary. Since every point between these two must be in the
Cantor set, points of the form xlx2. • xnl... must be in the set, contradicting
the fact that the Cantor set contains no points with ls in their ternary
expansion. This proves that the Cantor set contains no intervals, thus its
only "connected" subsets are single points. (We discuss the concept of
connectedness in detail in the chapter on Julia sets). This illustrates another
property of the Cantor set: it is totally disconnected.
Now a great paradox results. We have removed the entire length of the
interval. The only points remaining are the points of the form x = .xlx2x3 ...
where xi e (0, 2). Now we form the function f(x) = y where yi = xi/2. The
set of y values is the set of all strings of Os and is. The cardinality of this set
17
is the same as the cardinality of [0, 1] since there is a one-to-one
correspondence with the binary expansion of the points of the unit interval!
Thus, the cardinality of a set does not tell the whole story of "how many"
points are actually there from the point of view of length (in the one-
dimensional case).
There are other versions of Cantor sets resulting from similar
constructions. For instance, remove instead the middle fourth, fifth, or some
other fixed (1/N) length at each step. We can also remove from the unit
interval an open middle segment of length c/3k at the kth stage where
0 < a < 1 is fixed. Then we can show the total length removed is a. These
are frequently called "fat" Cantor sets, because the total length removed from
the unit interval is now less than one. These sets can also be described in
terms of a base N expansion, from which it can be shown that a fat Cantor set
contains no intervals.
18
Ill. FRACTALS
A. THE SETTING FOR FRACTALS
Fractal geometry involves the study of certain subsets of metric spaces. It
can be viewed as an extension of Euclidean geometry and is frequently used to
describe objects that occur in nature, such as crystals, plants, clouds, and
geological formations (see, for example, Cherbit (1991)). Fractals have been
applied to computer graphics to store information efficiently, examples of
which can be found in Barnsley (1988). Moreover, the use of fractals to study
real-world phenomena has provided a new way of analyzing the world.
Finally, as we shall see in Chapter IV, a primary use of fractals is to classify
and analyze chaotic dynamical systems.
The primary references for this chapter are Barnsley (1988) and Falconer
(1990). While the presentation most closely follows that of Barnsley, a more
rigorous mathematical development of most of the results can be found in
Falconer. Additionally, the article by Harrison (1989) covers much of the
same material, while many of the examples presented here are from Cherbit
(1991).
Much of the current literature differs in the precise definition of a fractal,
so our approach is to develop the setting for the space in which fractals exist.
Then we provide many examples of fractals in that setting. This app roach
provides a good initial understanding of fractals without the expense
involved in achieving a thorough understanding of their elusive definition.
19
A useful, although incomplete, definition of a fractal is that "a fractal is a
fixed point of a certain kind of transformation on the space (M(X), h(d))."
Thus, we must first define the set {(X) and the distance function h(d).
(*E(X), h(d)) is a metric space obtained from a complete metric space (X, d),
where the points in L(X) are dosed, bounded, nonempty subsets of X, and
where h(d) is a distance function based on the metric d, (which we define
shortly). The points in L(X) are called the "compact" subsets of X. To avoid
the necessity of employing concepts from advanced calculus, we consider only
Euclidean spaces where the compact sets are the dosed and bounded subsets.
Consider the Euclidean plane (R2, dE). We denote by WL(R 2) the
collection of all dosed and bounded subsets of R2, excluding the empty set.
Hence, the dosed unit square [0, 1]X[O, 1] belongs to W(R2), as does the origin;
however, the interval (0, 1)X{0) is not an element in WL(R2) since it is not a
dosed subset of R2. Clearly, the union of any two elements in 3,(R2) again
belongs to K(R2). The intersection of two elements in W(R2) is not
necessarily an element of W(R2) as the intersection may be empty.
In order to create a metric space out of the set KL(R 2), a distance function
(metric) is required that relates any two elements of iL(R 2) to a nonnegative
real number. To this end the concept of dilation is helpful. Given any dosed
subset A of R2 and e > 0, the e-dilation of A is defined to be the set of all
points x in R2 such that the smallest Euclidean distance between x and any
point in the set A is less than or equal to e; i.e., the e-dilation of A is the set
{x: dE(x, A) 5 e). For example, the e-dilation of the origin is simply the dosed
disc with radius E. The e-dilation of the unit square (0, 1]X[0, 1] with e = 1/2
is shown in Figure 3.1. The points of the square also belong to its e-dilation.
20
Yt
THEUNIT
SQUARE
Figure 3.1 The c-dilation of the unit square with e = 1/2.
We now define the distance from an element A in K(R2) to an element
B in K(R2) as the number dh(A, B) representing the smallest e such that
every point in B is covered by the e-dilation of the element A. In other
words, dh(A, B) = min(e: y e (the e-dilation of A) V y e B). Clearly, dh
exhibits the second and third properties of a metric from the original
definition. However, as shown in Figure 3.2, it is not symmetric since
dh(A, B) * dh(B, A) in general. To remedy this difficulty, we define h(A, B) =
max(dh(A, B), dh(B, A)). The number h(A, B) does satisfy the properties of a
metric for points in the set ;(R2), and is referred to as the Hausdorff distance.
Now the set IL(R 2) together with the metric h(A, B) is a metric space. It is
denoted by (M(R2), h(d)), or for purposes of simplicity just ILE, and we refer to
it as the Hausdorff-metric space. We remark that if the metric space (X, d) is
complete, then the associated Hausdorff-metric space MM(X), h(d)) is also
complete. In particular, IL is complete since X = R2 is complete.
21
d AB ) B d h (B, A)
Figure 3.2 The distances dh(AB) and dh(BA) are not equal in general.
B. CONTRACTION MAPPINGS
In order to generate fractals geometrically we need to be familiar with the
concept of contraction mappings. The results of this section are presented in
the setting of a general metric space. However, we concentrate our efforts on
three main spaces: the real line with the normal distance function, the
Euclidean plane, and the Hausdorff-metric space (W(R 2), h(dE)).
A mapping f: X--X is said to be acontraction mapping if there exists a
constant 0 < s < 1 such that d(f(x), f(y)) < s(d(x, y)) for all x, y e X. The
number s is called the contractivity factor for f. We prove shortly that a
contraction mapping is always continuous. Under a contraction, any two
points in the space that begin a distance D apart will be moved to within a
distance sD of each other. A key result concerning contraction mappings,
22
"and one which is critical when constructing fractals as subsets of the
Hausdorff-metric space K, is discussed next.
THE CONTRACTION MAPPING THEOREM.
If f: X--X is a contraction defined on a complete metric space (X, d), then f
has a unique fixed point xf e X. Furthermore, for any point x e X, the
sequence fn(x) converges to xf.
To prove this we first need the following results.
THEOREM. If f is a contraction, then it is continuous.
PROOF. If we let s be the contractivity factor of f, and if e > 0 is given, then
choosing 8 = e/s yields d(x, y) < 8 * d(f(x), f(y)) S s(d(x, y)) < s(e/s) = e. This
shows that f is continuous.
LEMMA. If f is a contraction with contractivity factor s, then for a fixed x
and m < n, d(fn(x), fMo(x)) s stud(x, fnTnl(x)).
PROOF. The proof follows immediately from the contractivity factor s, and
the principle of mathematical induction.
PROOF OF THE CONTRACTION MAPPING THEOREM.
Let x0 be an arbitrary point in the complete metric space (X, d), and let
f: X-4X be a contraction mapping such that xj = f(xo), x2 = f(xi) = f2(xo), and in
general, Xn = f(xn-l) = ff(x0). For positive integers m and n such that m < n,
we have
23
d(xn, xn) = d(fm(xo), fP(xo)) = d(fMXxo), fM(fI-m(xo)))
< smd(xO, fI-m(xo)) = smd(xo, xr-m)
<: sm[d(xo, xj) + d(xi, x2) +. .. + d(xnk-1, xn-m)]
(by the triangle inequality of d)
< smd(xo, xi)[1 + s + s2+.. .+ sn-D- 1 ]
(by applying the lemma to each term)
< [sm/(1 - s)ld(xo, xj) (by the geometric series).
Since s < 1, sin-4 0, hence d(xm, xn)--O. Since X is a complete metric space,
xn--xf for some xf e X, so the sequence (fn(xo)) converges to the point xf for
any xo in the space. To show that xf is a fixed point: since f is a continuous
function, f(xf) = f(lim Xn) = limr f(xn) = limr Xn+l = xf. Therefore, xf is a fixed
point of f.
To show that xf is the only fixed point of f, assume that y is also a fixed
point. Then d(xf, y) = d(f(xf), f(y)) < sd(xf, y). Since s < 1 and d a 0, this
implies that d(xf, y) - 0, so xf = y, completing the proof.
The importance of the contraction mapping theorem will become
apparent when fractals are created through iterated function systems. When
we find a point in the space that is a fixed point under a contraction mapping,
we then know it is unique (hence we will refer to "the" fixed point of the
mapping). Furthermore, we know that any initial point converges to the
fixed point under iteration of f.
24
C. AFFINE TRANSFORMATIONS OF THE PLANE
Affine mappings were defined earlier as functions of the form
fWx) = ax + b. In R2, we change our notation slightly and write an affine
transformation as w(x) = Ax + t, where x is a point in R2 (a column two-
vector), A is a 2X2 matrix, and t is a (fixed) vector in R2. We frequently
use the form
w(x) =w[iJ2=[a b] J÷:=Ax+t.x]', + [el ,
The mapping w(x) = Ax is called a linear transformation, and it takes any
parallelogram with one vertex at the origin into another parallelogram with a
vertex at the origin, provided that the determinant of A is not zero (that is,
A is nonsingular). Notice that the origin remains fixed under any linear
transformation. A result from linear algebra is that the matrix A can be
written in the form
[a bi r cosOl -r2sin92 ] 1LC d][ r1sinO1 r2cosD2 J
where the point (a, c) has polar coordinates (ri, 01) and the point (b, d) has
polar coordinates (r2, 02 + x/2). The transformation w(x) = Ax "deforms"
the space R2 relative to the origin. By deform, we refer to the result from
linear algebra that any linear transformation can be expressed as a succession
of shears, compressions, expansions, and rotations of the space. The result of
adding the vector t to a point in R2 shifts that point by the magnitude and
direction of t, so an affine transformation of the plane can be thought of as a
25
deformation (shearing, shrinking/stretching and rotating) followed by a
translation. Figure 3.3 shows the result of the affine transformation
wjjiJ,=[ 2 _O][x2,]+[-2]
onthetriangle T withvertices A = (0, 1), B = (1, 1), and C = (1, 0), where A',
B', and C are the images of A, B, and C under w, respectively.
x x2,
AtA1/2
0 1 x -2 -1 0 1
Figure 33 An affine transformation of a triangle.
We now consider several special affine transformations. From the
representation in polar coordinates Eq. (1) above, if ri = r2 = r, and 01 02= 0,
we say the transformation is a similitude, and write it in the form
SLrsin0 rcosOL2
or,
[:lr.o :si,,"lrxin l+[e{X2] [rsin0 -rcoseJ[ x2 J1 L
26
"A similitude scales any image in R2 by the same factor r in each component
direction, rotates it by the angle 6, and then translates it by the magnitude and
direction of the vector t. Recall from linear algebra that the transformation
[cosO -sinsinO cosOJ
is a rotation, and the transformation
is a reflection across the x-axis in R2. Both of these transformations behave as
one would expect, based on their names. Figure 3.4 shows the transformation
[] [cos'z/2) -sin~x/2) ][x] (2)LJ sin(r/2) cos(7i'2 jX2
of the unit square which is a rotation with 0 = x/2- Figure 3.5 shows the
transformation
W[2]= 0,][:21 '](3of the unit square which is a reflection about the x axis. Being familiar with
both reflections and rotations is very useful when creating fractals.
The action of w(x) = Ax + t can be determined by evaluating its effect on any
point (xj, x2) in the plane. However, it is sometimes more useful to
determine which affine transformation has caused a particular (observed)
change to a given set of points. The beauty of affine transformations is that
they are uniquely determined by their action on any three non-collinear
27
2 x2
C D' B'1
A C' A'0 11 1 x 1Ox 1
Figure 3.4 A rotation of the unit square by the transformation in Eq. (2).
X2
0 __ x1C D -N' B'
A E C' D'S1O 1
Figure 3.5 A reflection of the unit square by the transformation in Eq. (3).
points in the plane. To see this, start with any three points (xi, x2), (yi, y2),
and (zl, z2) and note their movement under an (unknown) affine
transformation to the new points (xj', x2'), (y0', y2'), and (zl', z2'). The
coefficients a, b, c, d, e, and f of the transformation can be determined from
the system of linear equations
xla + x2b + e = Xl',
yja + y2b + e = yI',
zla + z2b + e = zl',
28
x1 c + x2d + f = x2 ,
yic + y2d + f = y2',
zlc + z2d + f = z2'.
Because the points are non-collinear, the matrix of the coefficients is
nonsingular and the system has a unique solution. An example follows.
EXAMPLE. Using the example in Figure 3.3, the transformation on the
triangle transforming the points (1, 0), (1, 1), and (0, 1) to (-2, 1/2), (0, 1/2),
and (0, 1) respectively yields the following system of equations:
(0)a + (1)b + e = -2,
()a + (1)b + e = 0,
(l)a + (O)b + e = 0,
(O)c + (1)d + f = 1/2,
(1)c + (1)d + f = 1/2,
(1)c + (0)d + f = I.
Solving for the coefficients a through f yields a = 2, b = ,c0, d =-1/2,
e = -2, and f = 1, hence, the affine transformation is given by
w_,,2][x2X 0 IIXI
D. CONTRACTION MAPPINGS OF THE SPACE X00
Given n continuous maps of a metric space wl, w2,..., Wn: X-+X, we
construct a map W of the associated space 9 by considering
W(x) = wl(x) U w2(x) U... U wn(x),
29
• . . , i ial I I IM M = I l
"where x is any point in the space R, i.e., x is a closed, bounded, nonempty
subset of the original metric space. For x e K, we define Wi(X) = {wi(y): y e x
c X). We state here that the mapping W maps 1C to itself. This fact is based
on two important results from advanced calculus, which we state but do not
prove. The first result is that continuous images of compact sets are compact,
so that each wi maps K to itself. The second is that the finite union of
compact sets is compact. Thus, W maps K to itself, as claimed. For a simple
illustration of this concept, let n = 2 and X = R, and consider the maps on R
defined by wl(x) = x and w2(x) = 2x. If we start with the singleton compact
set x = (1), we have W(1) = wi({l)) u w2((1)) = (1) u (2) = (1, 2).
We next iterate the map W. The set of points W"(x) grows quite rapidly
as n increases. To see this, consider the same example used above with n = 2.
Then W2(x) = W(W(x)) = W(wj(x) u w2(x)) = wl(wl(x) u w2(x)) u w2(wl(x) v
W2(X)) = WI(WI(x)) U wl(w2(x)) U W2(Wl(X)) V W2(W2(X)). Using wl and W2
as above, and again starting with x = (1), we have W2({1}) W(1, 2)) =
(1, 2) u (2,4) = (1, 2,4). Similarly, W3 (1)) = (1, 2,4) u (2,4,8) = (1, 2,4, 8).
Note that wj(w2(x)) = w2(wl(x)) because wl and w2 are linear
transformations on R, which commute. For notational convenience, we
shall not distinguish between the notation x = 1 and x = (1) for the space
K(R) includes sets of singleton points.
We use the abbreviated notation wi(w1(x)) = Wij(x) etc. Then,
W1 = W1 u w2; W2 = WI1 Uw12 U w21 U w22, and W3 = WllIl Uw112 Uw121 U
W122 U W211 U W212 V W221 u w222. These subscripts may look familiar. We
are building, in this way, an iterated function system in one-to-one
correspondence with the code space 12. To make this clear, we let each I in
30
"our subscripts correspond to the 0 in code space, and let each 2 in our
subscripts correspond to the 1 in code space. Then, the iterate Wijk... is in
direct correspondence with the point i-1, i-i, k-1,... in Y,2. We could have
emphasized this more had we named our contractions wo and wi.
However, we save the designation wo for a specific use later on. Similarly, if
W is the union of n mappings, then the subscripts are analogous to the
points in In after infinite iteration. This concept proves quite useful when
we identify points on fractals through an addressing scheme.
Notice that the infinite iteration W"(1) in our simple example produces
the set (1, 2, 4, 8, 16,...). While the dynamics of this particular iterated
function system are not very exciting, when each wi is a contraction with
associated contractivity factor si the results do become much more
interesting, as we discuss next.
To discuss contraction mappings on the space 1E, we need the following
two theorems.
THEOREM. If a mapping on a metric space w. X--X is a contraction with
contractivity factor s, then w- L(X)--§L(X) is also a contraction with contrac-
tivity factor s.
PROOF. Let B, C e 1E. Then
dh(w(B), w(C)) = min(e{ y e (the e-dilation of w(B)) V y e w(C)}
< minise: y e (the e-dilation of B) V y reC}
=s min(e: y e (the e-dilation of B) V y e CI
=s dh(B, C).
Similarly, dh(w(C), w(B)) •s dh(C, B).
31
"Therefore, h(w(B), w(C)) < s h(B, C), completing the proof.
THEOREM. Let wl, w2, , wn be contractions on the space Ii with
contractivity factors sj, s2, ... , sn. Define W: It-&# by
W(B) = wl(B) u w2(B) u... u wn(B)
for each B c 9. Then W is a contraction mapping on 91 with contractivity
factor s = maxfsi: i = 1, 2,..., ).
PROOF. We prove this for the case n = 2. The rest follows by induction.
Let B, Ce H. Then
H(W(B), W(C)) = h(wI(B) u w2(B), wj(C) U w2(C))
< max(h(wl(B), wI(C)), h(w2(B), w2(C)))
< maxfslh(B, C), s2h(B, C)) = sh(B, C).
A simple exercise shows why the first inequality holds. This completes the
proof.
Figure 3.6 shows two contractions on the unit square in R2, wi contracts by
1/2 in the xI direction, and w2 contracts by 1/4 in the x2 direction. Their
union has a contractivity factor of 1/2 by the previous theorem.
Since W is a contraction on the complete metric space K(X) we know
that it has a unique fixed point in K00, which is a dosed, bounded, non-
empty subset of the space X. By the Contraction Mapping Theorem, all points
in KL tend to this fixed point under infinite iteration of W. This unique
fixed point is called the attractor of W, and frequently exhibits very interesting
properties.
32
x 2 x 21 1
S Wl(S)
x 1 xx 1 x1 1
w2 (S) W1 (W2 (S)) =
W 2(W1 (S))
x 1 1
Figure 3.6 The union of two contractions of the unit square.
E. CREATING FRACTALS THROUGH ITERATED FUNCTION SYSTEMS
1. The Cantor Set
We are now prepared to construct fractals using affine transfor-
mations and the Contraction Mapping Theorem. We start with the simplest
setting, namely the unit interval, and construct a fractal with which we are
already familiar.
Consider the union of two affine transformations of the real line R,
given by W(x) = wl(x) u w2(x), where wl(x) = x/3, and w2(x) = x/3 + 2/3. If
we first restrict our attention to the unit interval [0, 11, then under the first
iteration of wi the unit interval is shrunk by 1/3 towards the origin such
that [0, 1]-+[0, 1/31. Under the first iteration of w2 the unit interval is shrunk
by 1/3 and then translated to the right by 2/3 so that [0, 1]-+[2/3, 1]. Note
33
that the effect of W on [0, 1] is the removal of the open middle third of the
unit interval, the first step in constructing the Cantor "middle thirds" set.
Under the second iteration W2, the mappings wI and w2 now act on
the subsets [0, 1/31 and [2/3, 1] of the unit interval, so that W2[0, 1] = wlO[0, 1] u
W1210, 1] u W2110, 1] uw[220, 1]. This can be written as wj[0, 1/3] u wj[2/3, 1] u
w2[0,1/3] u w2[2/3, 11. Since wj[0, 1/31 = [0,1/91, wj12/3,1] = [2/9,1/31,
w2[0,1/31 = [2/3,7/91, and w2[2/3, 11 = [8/9,11, we obtain W2[0, 11 =
[0,1/91 u [2/9,1/31 u [2/3,7/91 u [8/9,11. This procedure has the same effect as
removing the open middle third of each interval [0, 1/3] and [2/3, 1].
Using a similar analysis, W3[0, 11 again removes the open middle
third of each of the four intervals produced from W2, and continuing in this
manner results in the construction of the Cantor middle-thirds set (see Figure
3.7).
0
Wl1O,.11 w2 [O, 1]
W 1 W 1 2 W2 1 W2 2
will w1, 1 2 WI 2 1 W12 2 211 w212__ w221 w222
Figure 3.7 Creating the Cantor set through an iterated function system (IFS).
The above construction is perhaps the simplest example of a fractal.
Further examination reveals that this set has some interesting properties.
First, consider the set of points that are fixed under the map W. Note that the
34
.fixed point of wl is 0, since (1/3)(0) = 0, and that the fixed point of w2 is 1,
since (1/3)(1) + 2/3 = 1. Since each of these mappings is a contraction of the
real line, each has a unique fixed point by the contraction mapping theorem.
However, under the union of these maps W = wI U W2, the endpoints of the
intervals we constructed, 1/3,2/3,1/9,2/9,7/9,8/9,1/27,..., also remain in
the set after each iteration of W. While we restricted our attention to the
unit interval, we note that each contraction wi is defined on the real line and
their union is a contraction on the space W(R). Hence, if we can find a closed,
bounded subset of the real line whose image under W is itself, we know it is
the unique fixed point of the mapping W. The classical Cantor set is such a
point, because taking the contraction W of the Cantor set again yields the
Cantor set. To see this, recall that in our construction of the Cantor set each
subinterval contains a smaller copy of the whole set because of the infinite
iteration we used. Likewise, since W maps the unit interval to [0, 1/3] and
[2/3, 11, each subinterval also contains a smaller copy of the whole interval.
Since the Cantor set is the fixed point of this iterated function system (IFS), we
refer to it as the attractor of W.
This illustrates a property of fractals known as self-similarity. If we
examine the interval [0, 1/3] in the construction of the Cantor set, we see that
it is a scaled copy of the original unit interval. Likewise, if we examine any
subinterval of the unit interval, we observe the same result. This is a unique
property of the infinite iteration scheme used to construct the Cantor set.
Similarly it is the result of iterating W an infinite number of times. This
self-similarity characteristic of fractals is very useful in applications to
computer graphics.
35
Because it was known ahead of time that the attractor of W would be
contained entirely within the unit interval, to simplify the discussion we
based our previous analysis on mappings of the unit interval W: [0, 11--[0, 11.
Now consider the result of mapping a different compact (i.e., dosed and
bounded) subset of the real line. From the Contraction Mapping Theorem,
no matter which nonempty compact subset of the real line we begin with,
infinite iteration of W always produces the Cantor set. Although we proved
this principle formally, it certainly is not intuitively obvious. In the next
section we illustrate this idea with an example in R2, where the geometry can
be better appreciated.
The question now arises, what happens to the entire real line under
iteration by W? It is dear that W(-"o, @0) = (-", oo), which appears to be a fixed
point of W. However, the set (-w., e) is not in the space M(R) since points
in the space *(R) are dosed and bounded subsets of R, but (-.,-.) is not
bounded.
This thorough examination of the Cantor set as a fractal produced by
the iterated function system W provides the basis for similar results in
higher dimensions. The principles remain unchanged and adding a
dimension yields ever more interesting results.
2. Fractals in Two Dimensions
We now lift the setting from the real line to the Euclidean plane R2
and the space (W(R2), h(d)) associated with it. Consider the following three
contractions of the plane:
36
W 32]=[ ][ J [ +[ 112[X 0 1 r/ ,W3[L2']-[" o ][X2.]+[ 0•
Let W = wl u w2 v w3. Then W is a contraction on K(R2), and has a unique
fixed point, or attractor. To find the attractor, consider the action of W on
the unit right triangle T with vertices (0, 0), (1, 0), and (0, 1). Since T is a
dosed and bounded, nonempty subset of R2, it is a point in K(R 2). Each of
the maps wl, w2, and w3, shrinks any point in R2 towards the origin by a
factor of 1/2 in the directions of each coordinate axis. The map w2 shifts
each point 1/2 unit to the right as well, while w3 shifts the result 1/2 unit
upwards.
Figure 3.8 shows T and the result of WM. Notice that the result of WM is
to remove a "middle third" triangle from the original triangle T. Again,
iterating T under W yields the results shown in Figure 3.9, displaying the
remaining triangles from their previous iterations with the middle thirds
removed.
Continuing to iterate W, we see that the fixed point of W is
obtained by continually removing the middle third of each subtriangle from
the original triangle T. This attractor is shown in Figure 3.10 and is called the
Sierpinski triangle (or Sierpinski gasket).
37
.nnMn iI
x x2
The Unit Triangle T 1 W(T)
Figure 3.8 The effect of W on the unit triangle.
X A
1N
x
Figure 3.9 The effect of W2 on the unit triangle.
The same results observed for the Cantor set in one dimension apply
to the Sierpinski triangle in two dimensions. The Sierpinski triangle has
infinite detail, and is self similar: each subtriangle contains a scaled copy of
the original unit triangle. Furthermore, while we used the unit right triangle
38
"T as the starting point, by the Contraction Mapping Theorem any point in the
space Wfl(R2) converges to the Sierpinski triangle under infinite iteration of
W. Figure 3.11 shows how the unit square with infinite iteration under W
also yields the Sierpinski triangle.
Figure 3.12 shows how the point (1, 1) starts away from the attractor,
but under iteration of W eventually converges to points on the attractor.
Figure 3.13 shows the effect of W6(S) on the unit square.
Figure 3.10 The Sierpinski triangle.
39
x22 1
~ W(S)
0 0
A S) w3Ss)
0 x 0 x1 11
Figure 3.11 Creating the Sierpinski triangle by iterating the unit square.
40
& x x21 2 (1, 1) 1 2
W3,,W(S)
THE UNIT SQUARE
L
0 "-x 0 x1 01
Figure 3.12 The orbit of the point (1, 1) under iteration of W.
Figure 3.13 The effect of W6(S) on the unit square.
41
* , 3. Condensation Sets
So far we have only considered examples of contractions where
0 < s < 1. We will now consider contractions where the contractivity factor is
s = 0. Such contractions are called condensations, and in one dimension this
corresponds to fRx) = c where c is a constant It is clear that no matter what
two points we start with, the distance between their first iterates under f is 0,
since c - c = 0, so that f is a contraction.
In the space {(X), wo: H(X)-+H(X) is a condensation if wo(x) = B for
all x in *E(X), where B is some fixed point in 1(X). If we take a contraction
mapping of the space K, W = wi u...u wn, and form the union of it with wo,
where wo is a condensation such that W'= W u wo, then W' will be a
contraction with the same contractivity factor as W. To see this, consider that
the contractivity factor of W is s = max(s1, s2,., SIn, and that the
contractivity factor of W is s' = max(O, sl, s2,..., sn} = s.
Although the contractivity factors of the maps W and W are the
same, their dynamics are quite different. Since every iteration of W maps
points to the point B, the attractor of this iterated function system can be
thought of as the infinite set of every iteration of B under W.
As an example, consider the one-dimensional case with the two
functions wi(x). = x/2 and wo(x) = 1 on the real line. The map wl is a
contraction, and wo is a condensation, and their fixed points are 0 and I
respectively. If we start with a point on the real line, for example 0, and
iterate it under W = wo u wl, we see that W(0) = (0, 1), W2(0) = [0,1,1/2),
W3(0) = 0,1,1/2,1/4), and Wn(O) = (0, 1, 1/2,1/4,...,1/2'I). Clearly, this
42
-iterated function system generates an attractor which consists of every iterate
of the condensation set (1) under the contraction wi(x) = x/2.
4. A Fractal Tree
We now use the concept of condensation sets to construct a fractal
image. We first let the set B = (xj, x2): -0.1 s xi r 0.1; 0: O x2 5 1). In other
words, B is the filled rectangle with height I and width .2 with base centered
at the origin. Now define the transformations
wO[xi]=B,
[2]=[7csn4 -.75sim(xI4) I[x2,] +01x .75sin(x/4) .75cos(r,4)JX 2
W[X2ii-I .75cos(-I4) -.75sin(,r14)][x,].[O0
X .75osi(-Wr4) -.75os(-u./4) x+ [The transformations wi and w2 shrink any image by three-fourths in the
direction of both axes, w1 rotates it 45 clockwise and w2 rotates it 45"
counterclockwise, and then both transformations shift the image up one unit.
The transformation wo is a condensation that maps any point to the set B.
Ifwelet W=wouwluw2, and we iterate the origin under W, we
see that W(0, 0) yields the set B, W2 yields the set B together with the two
condensed, rotated, shifted copies of B, and W3 yields the set B together
with two condensed, rotated, and shifted copies of the image of W2. If we
continue iterating, we see that every iteration yields the set B together with
43
two condensed, rotated, and shifted copies of the previous iteration. The first
seven iterations of the origin under W are shown in Figure 3.14, which is
the beginning of a fractal "tree."
Figure 3.14 A fractal tree.
44
Another way of generating a fractal tree is through an iterated
function system without condensation. If we consider the transformations
w x2j] = -20 x2] [.045
W[2 .1 sin(XI4) .25COS(W/4 .5]
W3 [xi: [ .lcos(-x/4) -.25sin(-/4)lr,2J.,[.51
3 [] = si([ /4 .25cos(-x/4) xL~2 .5[and we consider the effect of W = wl v w2 u w3 on the unit square, we see
the result shown in Figure 3.15. If we continue to iterate W, we produce
the attractor of this iterated function system, which is a fractal tree with a
similar shape to the one in Figure 3.14. However, unlike the tree with
condensation, the trunk and branches of this tree are not solid, in fact, each
contains a condensed copy of the entire attractor. These detailed images are
not only aesthetically pleasing, but they lead to an important application of
fractals. To store these images in a computer using traditional means would
require hundreds of lines of code; however, we have used just 18 numbers
(three two-by-two matrices and three two-vectors) to record all of the
information required to produce these images. This efficient way of recording
information has led to important applications of fractal geometry, as the next
section shows.
45
"x 2
THE UNITSQUARE "S"
(S)
0 1 X
Figure 3.15 The action of W on the unit square.
F. APPLICATIONS OF FRACrALS TO COMPUT GRAPIHCS
Pictures that are infinitely detailed and self similar can be stored in a
computer with very small amounts of information. For example, in the case
of the fractal tree in the last section, only 18 numbers were required. But what
if an image is not self similar? Is it still possible to use contraction mappings
to store the information? The remarkable answer is "yes." Perhaps even
more surprising, this procedure can be achieved to any desired degree of
accuracy.
A formal description of using contractive iterated function systems to
store graphical information can be found in Barnsley (1988) or Falconer (1990).
Instead of presenting the proofs here, we give an intuitive treatment of the
technique through an example. Then we show how any desired degree of
accuracy can be attained.
46
Recall that affine transformations can have many different actions on a
set, including shearing, shrinking or stretching, rotating, and shifting.
Suppose we have an image we wish to reproduce; let that be our attractor.
We then search for affine transformations that could paste distorted copies of
the set back on top of itself to approximate the original image as closely as
possible. This is done through the "collage theorem" due to Barnsley. We
present an example here.
Consider the image in Figure 3.16. While this figure is not self similar,
we can approximate it by making smaller copies of it (through contractions)
and "pasting" them back on itself (through affine transformations) to cover,
Figure 3.16 A nonself-similar image.
47
as much as possible, the original figure. One such covering is shown in
Figure 3.17. The four corners of the original image and their movement
under the coverings are indicated by the letters A through D. Note that a
certain amount of overlapping of the covering is required, for if this were
just-touching or totally disconnected, the attractor would be too sparse to
represent the original image.
C 1C 1 A 2C D2A
3B 3D
1 1 2 B2D
112 2 DIA 5A 4A s 1B 3 3C B
Figure 3.17 Covering an image with transformed copies of itself.
Recalling that an affine transformation is uniquely determined by its
action on any three noncollinear points, we could easily determine the actual
transformations required to produce the seven mappings indicated in the
figure. Under repeated iteration of this iterated function system, we could
then reproduce the image in a similar way as that used to produce other
attractors, with only 42 pieces of data, since each affine transformation
requires six numbers.
48
But notice that the attractor of this iterated function system is not the
original image, since its boundaries are infinitely detailed, as opposed to the
solid boundaries in the original image (see Figure 3.18). This "fuzziness"
seems to contradict the earlier claim that we could approximate a computer
image to any degree of accuracy. But the above system used only seven
transformations. Noting that any computer image is displayed by a finite
number of pixels, which are either on or off, we could push this technique to
its extreme to reproduce exactly the original image. That is, we could
condense the entire set down to the size of one pixel, and then shift that pixel
through affine transformations to every pixel in the original image. This will
indeed give an exact reproduction of the original image. However, it would
require six numbers for every pixel in the image.
Figure 3.18 The attractor with a fractal-like boundary.
49
While this technique obviously defeats the concept of efficient storage of
information, it nevertheless shows that we can achieve any desired degree of
accuracy. Depending on the requirements of a given project, iterated function
systems and their attractors can frequently be used to achieve the desired
degree of accuracy much more efficiently than the traditional methods of
storing information.
So far we have claimed that only the information contained in the affine
transformations is needed to use this technique. In fact, we also require a
program to produce the image based on the input. However, these are quite
common and there are a variety of algorithms that produce the desired
results. Thus, in a situation where many different figures must be recorded, a
single program will suffice for creating the graphics once the transformations
are stored. It follows that this method is more efficient than many traditional
computer graphics techniques.
G. THE ADDRESSES OF POINTS ON FRACrALS
A useful technique in analyzing fractals is to address every point on an
attractor by the sequence of transformations that led to that point being in the
attractor. This leads to the classification of iterated function systems as being
"totally disconnected," "just touching," or "overlapping," and helps us
analyze chaotic dynamics on fractals in Chapter IV.
We will start with transformations of the real line to illustrate this
concept in the simplest setting possible. Consider the system wO(x) = x/3 and
w2(x) = x/3 + 2/3 (the reason for selecting these subscripts will soon become
apparent). We already know that repeated iteration of W = wo u w2 yields
50
"the Cantor middle-thirds set as an attractor. (Recall that there are no intervals
in the Cantor set, and that the intersection of wo and w2 applied to the unit
interval, or any subset thereof, is empty). Considering the iteration of W on
the unit interval, we now begin to address the points in its attractor.
Under the first iteration of W, the unit interval is mapped to the interval
[0,1/3] by wo and to [2/3,1] by w2. Hence, we begin the address of every
point in [0, 1/31 with 0, and we begin the address of every point in [2/3, 11
with 2. Under the second iteration of W, the interval [0, 1/31 is mapped to
[0,1/9] by wo and to [2/3,7/91 by w2, whereas the interval [2/3, 1] is
mapped to [2/9,1/31 by wo and to [8/9,11 by w2. Hence, the second number
in the address of all points in [0,1/9] and [2/3, 7/91 is 0, and the second
number in the address of all points in [2/9,1/31 and [8/9,11 is 2. Figure 3.19
shows the first three steps of this process, and the beginning of the resulting
addresses of every point in the Cantor set.
0 1
0 2
00 02 20 22
000 002 020 022 200 202 220 222
Figure 3.19 Addressing the attractor of an iterated function system.
It is dear that under infinite iteration, every point in the Cantor set has a
unique address consisting of an infinite string of Os and 2s. Note that the
51
origin has the address 000... and that 1 has the address 222... What we
have by the convenient selection of 0 and 2 as the subscripts is that every
point in the Cantor set is addressed by its ternary expansion, as described in
Chapter II.
The addressing scheme used here is not unique, as we could have also
chosen 0 and 1 as subscripts resulting in every point in the Cantor set
having an address from the code space £2. Again, note that every point in
£2 would be an address of a point in the Cantor set, and that any dynamics we
performed on the code space £2 could easily be applied to the Cantor set.
This one-to-one correspondence between code space and fractals is very useful
in the discussion of chaotic dynamics on attractors of iterated function
systems. (As an aside, we could have just as easily selected 1 and 2 as
subscripts to demonstrate further that an addressing scheme is not unique.
Nevertheless, all of these systems have a one-to-one correspondence between
them.)
Notice that every point in the Cantor set has a unique address associated
with it. This characteristic allows us to classify the iterated function system
W = wo u w2 as being totally disconnected. While the attracting set (the
Cantor set) is disconnected, we associate the classification "totally
disconnected" not with the attractor, but rather with the iterated function
system that produced it. The reason for this will become apparent when we
look at "just touching" and "overlapping" iterated function systems which
have the same set as their attractor.
Now consider the iterated function system W = wi u w2, where
wi(x) = x/2 and w2(x) = x/2 + 1/2. The unique fixed point (and hence the
52
' a l l l l Ii i
attractor) of this system is clearly the unit interval. So if we consider the
action of this system on [0, 1], we can build addresses for every point in that
interval. Starting with [0,1], we see that wl maps it to [0,1/21 so we assign
to that interval the first address of 1, and similarly points in [1/2, 1] have the
first address 2. Already we can see that there is some ambiguity with the
point 1/2, since it appears to be receiving two distinct addresses. Continuing,
we see that points in the interval [0, 1/41 have addresses starting with 11,
while points in the intervals [1/4,1/21, [1/2,3/41, and [3/4,11 have addresses
12, 21, and 22, respectively. Now the points 1/4 and 3/4 have been added
to the list of points with dual representations. Figure 3.20 shows the first
three iterations of this addressing scheme.
0 114 1/2 3/4 1
1 2
11 12 21 22
111 121 211 221-112-122 - 212-222
Figure 3.20 Addressing the attractor of a just-touching IFS.
Again, it is dear that every point in the unit interval receives an address,
but points of the form i/2n, i = 1, 2,3,..., 2n - 1 have dual representations
under this scheme. The set of points where the addresses touch, although
infinite, is countable (i.e., is in one-to-one correspondence with the natural
53
numbers) since at every step we add a finite number of points with dual
addresses. Also, notice that no intervals have dual addresses, since between
every pair of points with dual addresses, there are points with unique
addresses. Iterated function systems with this characteristic are classified as
just-touching. This fits the intuitive concept of this definition since the
results of iterating the attractor [0, 1] under w, and w2 results in two intervals
which "just touch" at one point.
Finally, we consider the system W = wi u w2 where wl(x) = 2x/3 and
w2(x) - 2x/3 + 1/3. Again, the attractor of this system is dearly [0,11, but
when we address points in the attractor, we encounter considerable
ambiguity. Under the first iteration of W applied to [0, 1], we see that the
interval [0,2/3] gets the address 1, while the interval [1/3,1] gets the address
2. Here an entire interval [1/3, 2/3] of points has a dual address after just the
first iteration. Moreover, at every step we add intervals with a similar
ambiguity. The first three steps in this addressing scheme are shown in
Figure 3.21. Notice that in order to remove the ambiguity of the overlapping
addresses, we have to lift" the unit interval into a second dimension as the
figure shows.
The ambiguity in this addressing system is much greater than for the just-
touching iterated function system. In fact, at every step an uncountable
number of points with multiple addresses is obtained. Here, the number of
points with multiple addresses far outnumbers the points with unique
addresses. What is more, for this system, the only points with unique
addresses are 0 and 1. This feature leads to the intuitive definition of an
overlapping iterated function system. Notice that each iteration of wl and
54
"w2 results in an interval where different addresses overlap. Because the
attractor of this system is the same as in the example of a just-touching
iterated function system, it should now be clear why the classifications
"totally disconnected," "just touching," and "overlapping" are applied to the
iterated function system rather then to the attractor itself.
0 1/3 2/3 1
12
12
111 12 121
122Ill 112. 12 122
211 21221222
Figure 3.21 Addressing the attractor of an overlapping IFS.
With the concept of addressing points on attractors and classifying
iterated function systems in a simple one-dimensional setting, let us now
turn to a two- dimensional example. Recall the iterated function system that
produced the Sierpinski triangle as an attractor:
[x2 0 1/2 I[x]Jjj
55
W3[J2'=[olnJ([X]+[ 01J,1
where W = wl u w2 u w3. Iterating the unit right triangle under W yields the
Sierpinski triangle by removing the "middle-thirds triangle" at every step.
Looking at just one iteration of W, we see that this is a just-touching iterated
function system: the image of wi intersects the image of w2 at (1/2, 0), the
images of w2 and w3 intersect at (1/2,1/2), and the images of wi and w3
intersect at (0, 1/2). An addressing scheme for the Sierpinski triangle is
shown in Figure 3.22.
33
3 3
Figure 3.22 An addressing scheme for the Sierpinski triangle.
As before, the vertices created at each step have two distinct
addresses. For example, the point (1/2,1/2) has the addresses 2333... and
3222.... The exceptions are the three original vertices which have the unique
56
-addresses 111..., 222...,and 333.... Again, we could have chosen the
addresses to consist of Os, is, and 2s, to create a correspondence between the
Sierpinski triangle and code space 73. However, it is just as convenient to
create a modified code space r3 to be (xlx2x3...: xi e (1, 2, 3)). The metric
associated with this space 1-3 is
d(x, y) = x . ,O
i=l 41
so 1'3 is dearly in one-to-one correspondence with the Sierpinski triangle.
A more difficult addressing problem occurs with an overlapping
iterated function system in two dimensions. Recall that in an overlapping
iterated function system of the real line, we had to "lift" the attractor into a
second dimension to uniquely determine addresses. Similarly, in two
dimensions, we must lift the attractor into a third dimension to address its
points uniquely. For example, consider the overlapping iterated function
system of the plane W = wl u w2 u w3, where
rWi Xi _20 ir..olWl[xJ=[Z/3 0Jx[J
W2[J-j" ]23[X]J+[01/3
X] _[2/3 0 ][x [l 0
W 0 2/3j[21.,j
The attractor for this iterated function system is the "filled"
Sierpinski triangle, or simply the unit right triangle. Figure 3.23 shows the
first iteration of W and how the iterates of wi, w2, and w3 overlap. Any
57
point along the xi-axis in the interval [1/3,2/3] has an address beginning
with a 1 or a 2, and entire regions of the plane have similar ambiguities.
3
lor3 2or3
1 2
1 or 2
Figure 3.23 Ambiguous addresses for the attractor of an overlapping IFS.
While points on the attractor always correspond to multiple
addresses, we would like to identify every possible address with a unique
point. This is useful, for example, when considering the dynamics of the shift
map on code space associated with the attractor of an iterated function system.
Developing such an addressing scheme in two dimensions for this attractor
would be extremely difficult. However, "lifting" the attractor into a third
dimension allows us to create an unambiguous address system for the
attractor. Figure 3.24 shows how the first iterate of W (from Figure 3.23) is
lifted into the third dimension. This removes the ambiguous regions from
Figure 3.23.
We now lift the entire attractor into the third dimension. To
accomplish this, cross the Euclidean plane with the unit interval [0, 1] which
is represented by its base four expansion so that every point in the interval is
represented by a number .xix2x3... where xi e (0, 1, 2, 3). By associating the
58
'address of each iterate of W with its base four expansion, we see that we get a
Cantor-type set construction along the vertical axis when viewed from the
side. Since we used the subscripts 1, 2, and 3 for the iterated function system,
every number in [0, 1] that does not have a 0 in its base four expansion will
have an iterate associated with it. The result of this process, taken to three
iterates, is shown in figure 3.25. Having an addressing scheme such as this to
locate a unique point proves quite useful when we study chaotic dynamical
systems on fractals in Chapter IV.
Figure 3.24 Lifting an attractor into the third dimension.
59
13
2
212
• 2302
Figure 3.25 Addressing the lifted attractor of an overlapping IFS.
H. FRACTAL DIMENSION
One of the most useful concepts in the application of fractal geometry is
that of fractal dimension. Fractal dimension provides a measure of the size
or "dimension" of an object, whether it is the attractor of an iterated function
system, a more familiar geometric figure, or something arrived at through
collection of real-world data. Fractal dimension has worked its way into
many fields such as physics, meteorology, aeronautical engineering, and
60
2,mli em a1im
oceanography, and we provide some examples of these applications shortly.
To begin we develop this concept mathematically.
The fractal dimension of an object is the assignment of a number to the
object that represents how much space the object takes up in its ambient
space. We are all familiar with objects in one, two, or three dimensions. That
an object may have a non-integer or fractional dimension seems
counterintuitive. Nevertheless, you will see that the definition of fractal
dimension supports completely our intuitive concept of one, two or three
dimensional objects. With this in mind, we give some preliminary
definitions that are needed to develop the concept of fractal dimension.
Ane -ball about a point xo in a metric space is the set of all points in the
space within distance e > 0 from the point xO. Conventional notation for an
e-ball is B(xO, e) = (x: d(x, xo) < e). Notice that by using the "!" sign in the
definition, the e-balls in our discussion are dosed. While this notation may
be slightly unfamiliar, the concept certainly is not. For if we take the real line
with the standard distance function as ow inetric space, then B(1, 1/2) is
simply the dosed interval [0.5, 1.51. Furthermore, if the metric space is the
Euclidean plane R2, then B((0, 0),1) is simply the closed unit disc centered at
the origin.
Given a closed bounded nonempty subset of a metric space, we want to
cover that subset with balls of specified radius e. In the case of the real line, to
cover the unit interval with balls of radius 1/4 requires at least two (centered
at 1/4 and 3/4). We could use any number of such balls if we allow
overlapping. We are primarily interested in covering a set with the smallest
number of balls possible, so we define the integer N(A, E) to be the smallest
61
"number of closed balls of radius e needed to cover the set A. Specifically, for
any set A in our metric space, we define N(A, e) to be the smallest positive
integer M such thatMu B(xn, E) =)A,
n=1
for some distinct set of points {xn: n = 1, 2,..., M) in the metric space.
The Heine-Borel Theorem (see an advanced calculus text) guarantees that
we are always able to cover a dosed bounded set with a finite number of
c-balls. Since every set of positive integers has a smallest member, the
number N(A, e) uniquely exists.
We are now prepared to define fractal dimension. If we let A be a point
in the space K(X) (i.e., A is a dosed bounded nonempty subset of the metric
space X) the quantity D is defined asD ffi ln(N(A, e))
£-,O In(1/•)
The number D, if it exists, is called the fractal dimension of A and is denoted
D(A).
Let us clarify this concept with some intuitive examples. First, consider a
point x in R2. No matter how small we choose e, we can always cover the
point x with a single ball of radius e, so that D(x) = lime..ln(N(x, e))/ln(1/e)
Slime-Oln(1)/ln(/I /) = 0. Hence, for any point x in R2, we see that its fractal
dimension is zero (which fits our intuitive concept of a point having zero
dimension).
Consider next the unit interval [0, 11 as a subset of the real line.
Recalling that in R, e-balls are the closed intervals of length 2e, we begin to
62
cover [0, 1] with e-balls. If we let e = 1/2,1/4,1/8,.. ,1/2n,. •then the
number of intervals required to cover [0, 1] are 1, 2, 4,... , and 2n-1, respec-
tively. Clearly, n-ý- implies e-40, so D([0, 1]) - lime_-oln(N(A, e))/ln(1/e) =
limn...ln(2n-1)/in(2n) = limn-.4 (n-1)ln2/nln2 = limn_.(n - 1)/n = 1. Hence,
the fractal dimension of a line segment is 1, which is consistent with our
ordinary concept of dimension.
In the previous example we were very specific in the way we let E-.0.
This example leads us to an equivalent definition of fractal dimension. In the
same setting as before, let en = rn for 0 < r < 1 and n = 1, 2, 3,. Then
D(A) = limn_ in(N(A, En))/ln(1/En).
To show that these definitions are equivalent, let f(e) = Max(en: en < e), and
assume that e < r. Then f(e) • e: f(e)/r, and N(A, f(e)) ý N(A, e) a
N(A, f(£)/r). Since, for x > 1, ln(x) is a positive increasing function, we haveIn(N(A, f(e)/r)) < ln(N(A, e)) l In(N(A, f(e)))
ln(1/f(e)) ln(I/E) In(r/f(l))
We can assume that N(A, e)--+o as e-*0, or the result follows immediately.
Considering the left side, and taking the limit as e-0O, since for some n,
f(e) = en, we have
lim4,In(N(A, f(E)fr))1 =im [ln(N(A, e,-i)).
a- ) ln(l/fe .i) J In(lI£e)
Similarly, the right side yields
EM ln(N(A, f(e)))i lim ln(N(A, PQ)
- l" -- -r - " - 63 l n (r/-
63
im[ ln(r.N(A, en)) 1 limrln(N(A, en)1SI In() I+ - I II I II i il' i iEi)
.The pinching theorem from calculus establishes the equivalence of these
definitions.
Notice that in R2 with the Euclidean metric, the e-balls always give
closed discs. Frequently we find it more convenient to use "boxes" or dosed
squares to cover our sets. For example, if we want to cover the unit square
[0, 1IX[O, 1], it would be much easier to calculate the required number of boxes
of a given size than to calculate the number of C-balls. It turns out that we can
obtain the same value of fractal dimension using closed boxes as we obtain
using C-balls.
First, consider a grid in R2. Figure 3.26 shows a covering of this grid by
dosed discs, and we see that if we were to refine our grid and let the radii of
the discs go to zero, then the number of boxes and the number of discs will
remain in one-to-one correspondence.
Figure 3.26 Covering a grid with dosed discs.
64
The result of placing finer and finer grids with side length 1/2n over a set
in R2, and letting Nn(A) be the number of grid squares that intersect the
object, gives a third equivalent definition of fractal dimension as follows:D ia) rln(Nn(A))]
D(A) n4 ln(2~) IActually several equivalent definitions of fractal dimension are used in
different applications, depending on which is the most convenient. All of the
definitions give the same result. Therefore, both theoretical and
experimental applications are quite easy to perform in many different
situations. Different examples of these definitions for Euclidean spaces are
A limln(N(A, e))C(A = ln(l/e)JI
where N(A, e) is
1. The smallest number of dosed balls of radius e that cover A (our first
definition);
2. The smallest number of boxes of side e that cover A;
3. The smallest number of sets of diameter of at most e that cover A
(where the diameter of a general set is the largest distance between all
pairs of points in the set); and
4. The largest number of disjoint balls of radius e with centers in A.
We have presented these definitions in the context of R and R2, but they
are easily extended to R3, where e-balls become closed spheres and e-boxes
become cubes. More generally, the same theory holds for Rn for any integer
n, although the applications are of a more theoretical nature.
65
Now that we have discussed equivalent definitions of fractal dimension,
let us use the most convenient one to compute the dimension of the closed
unit square [0, 11X[0, 11. Using squares with side length 1,1/2, 1/4,... 1/2n in
R2, we see that 1, 4,16,..., V4 of them are required to cover the unit square,
respectively. Then
D([o. lJX(0, 1]) = ii'jln(Nn(A))
we have
D([0, liX[O, l])= nI. I-4)- = - =l,,> 2."IL n(2R)J l. 0 f~
Hence, our definition of fractal dimension again supports our knowledge that
the closed unit square has dimension 2.
We now apply the definition of fractal dimension to the Cantor set and
the Sierpinski triangle. Intuitively, one might believe that, since these objects
lie in R and R2 respectively, they should have dimersions 1 and 2. But recall
that we removed a certain amount of length or area at each iteration for an
infinite number of iterations. Thus it might be plausible for these objects to
have a smaller dimension than that of the spaces in which they exist.
Consider first the Cantor set as a subset of the unit interval. Let the e-balls
in R be the intervale of length 1, 1/3, 1/9,..., 1/3n. One can easily see that
1, 2,4,..., 2n balls are required to cover the set, respectively. Hence, the
fractal dimension of the Cantor set is
D im [n(2)] lIm n ln2
66n n- -
66t
"so that the Cantor set has dimension of approximately .6309. We removed a
total length of 1 from the unit interval when constructing the Cantor set, but
still left an uncountable number of points. Thus intuitively we might expect
its dimension to be somewhere between 0 and 1. Fractal dimension
provides a measure of how many points actually remain in the set.
Moving next to the Sierpinski triangle, if we use squares of side length 1,
1/2,1/4,..., 1/2n, then 1, 3,9,...,3n squares are required to cover the set
(see Figure 3.27).
Length - 1 Length -1/2 Length -1/4
1 1 1
One Square Three Squares Nine Squares
Figure 3.27 Covering the Sierpinski triangle with grids of size 1/2n.
Hence, the dimension of the Sierpinski triangle is given by
Ds) [him (31)l=ur111 ",3 1nn-')- _t2= n.. j R =E
and the Sierpinsid triangle has dimension of approximately 1.585. Again,
considering the amount of area removed from the unit triangle, it seems
reasonable that the dimension should be between 1 and 2. It now becomes
clear why the term "fractal" was chosen to describe objects like the Cantor set
67
"and the Sierpinski triangle, as they have dimensions which are fractions of
whole numbers.
While we have shown how to determine the fractal dimension
theoretically in some very simple cases, we are not always able to make this
computation by just looking at the attractor geometrically. However, if the
iterated function system that created the attractor is known, the fractal
dimension of the attractor can be determined by analyzing the iterated
function system itself. We discuss this idea next, and state a general result.
If we know that W = wl u w2 u... u wn is the iterated function system
producing an attractor, and if each wj is a similitude with contractivity factor
si (as defined in Section ll3, in that it is a shrinking in each direction by the
factor r such that si = ri, followed by a rotation and a translation) then if the
iterated function system is totally disconnected or just touching, the attractor
has fractal dimension D(A) given by the unique solution of
•Isil D(A)= I,
i-I
where D(A) will be between 0 and the dimension of the ambient Euclidean
space. If the iterated function system is overlapping, then we only get a
bound for D(A) given by D(A) ! D, wheren D
D•1sil I' I.
i-I
Even if we do not know the exact iterated function system that produced
the attractor, we can measure the contraction of our set under each
similitude, and solve for a rough approximation of D(A). While the proof of
this result is quite complicated, (see Barnsley, 1988, p. 185) it does give a very
useful theoretical tool for determining the fractal dimension of an attractor.
68
To demonstrate its use on the Sierpinski triangle, which we know is created
with three similitudes with contractivity factor of 1/2, and knowing that the
Sierpinski triangle is just touching, we have (1/ 2)D + (1/2)D + (1/2)D = 1,
from which we get D = ln3/ln2, as expected. Unfortunately, the theoretical
determination of fractal dimension is not always possible, so we must
frequently use experimental methods in applications.
L EXPERIMENTAL DETERMINATION OF FRACTAL DIMENSION
Fractal dimension, as previously discussed, is perhaps the most useful
aspect of fractal geometry in real-world applications. It has permitted the
analysis of numerous natural phenomena that previously were inaccessible
to scientists because of a lack of tools and theory. In nature, nice standard
geometric shapes are primarily the exception rather than the rule; irregular,
fractal-like shapes are found everywhere. Consider, for instance, the shapes
of clouds, coastlines, geological formations, atmospheric phenomena, plants,
crystals, and microorganisms. None of these objects can be described by a
"nice" geometric shape, but fractal geometry has provided a new way of
analyzing them.
One way scientists have studied nature through fractal geometry is by
comparing the fractal dimension of an observed natural phenomenon to a
system with a known fractal dimension (whether the known system was
created in a laboratory, created mathematically, or is a natural phenomenon
that has previously been studied). Similar fractal dimensions are used to
support hypotheses about the similarity of the two systems. Thus, the
69
experimental determination of fractal dimension has become quite
important.
Currently, there is no definite theory for determining fractal dimension
experimentally. However, what follows is one of the most widely used
methods. When real-world data are gathered experimentally, they can
usually be plotted as a subset of one of the Euclidean spaces R, R2, or R3.
Using e-balls of varying radii, one can "cover" the data with the smallest
number of balls possible, obtaining values for e and N(A, e). This procedure
is not as simple as it may at first seem, because though we know when a set of
points is covered, we can never be sure that the smallest number of balls
possible has been used. Nevertheless, with the aid of computers, a very good
approximation of N(A, e) for a given e can frequently be obtained.
In R2, one way of accomplishing this is to superimpose a mesh of grid
size e over the data points, and count the number of squares that contain
data points; this procedure has an obvious analogy in R3. By varying the
value of e, pairs of values for e and N(A, e) can be obtained. Plotting the
logs of these values against each other over a range of different values of e
should yield (approximately) a straight line from which the fractal dimension
of the set can be determined.
There are some drawbacks to this method. The first problem is that the
theory we have developed for fractal dimension applies only to closed,
bounded subsets of some Euclidean space. The real world, on the other hand,
does not appear to be closed or bounded. Nevertheless, this problem is easily
overcome by the fact that once we "measure" the world (i.e., collect a set of
data points experimentally) we reduce the real-world phenomenon to a
70
'cdosed, bounded set (since we can only collect a finite number of pieces of
information). Since all of our observations of nature are reduced to a finite
set, the dosed and bounded assumption fits nicely with our view of the real
world.
There is, however, another problem with the experimental method; a
problem which is not so easily dismissed. Recall that the definition of fractal
dimension invokes the limit as e approaches zero. But this limiting process
cannot be reproduced experimentally. For the fractals created through
infinite iterations of function systems, the attractors are known down to the
most infinite geometric detail. Moreover, they are completely self-similar at
every stage of iteration. However, we do not have this luxury in the real
world. We only obtain values of e and N(A, e) for a finite set of nonzero e
values, with no guarantee that the observed behavior will continue as e
approaches 0. For example, consider a piece of coral which appears to be
similar in construction to a fractal tree. On the macroscopic level we find
numerous self-similar aspects of the geometric shape of the coral. However,
as smaller and smaller values of e are taken, we find that on the cellular
level there is no similarity between the shapes of the cells and the shape of
the coral. Further analysis reveals that the cell walls themselves even have a
different fractal dimension. Moreover, this problem could perpetuate
endlessly as we continue to decrease e.
As another example of this problem, consider the dimension of a ball of
string. From a distance, the ball appears to occupy three dimensions in space,
so we might conclude its dimension is 3. However, as we move closer to the
ball, we see it not as a solid object but as a long string wound around itself, so
71
we might conclude it has only one dimension. As we get closer still, we see
the thickness of the string, and again we conclude it is three-dimensional, but
further inspection shows many one-dimensional fibers wound together to
make the string. These examples demonstrate the inherent problem with
calculating fractal dimension over different ranges of e, and we have not yet
taken e anywhere near 0.
Since we are plotting e and N(A, e) only over a finite range of e values,
any conclusions drawn about fractal dimension must rely on the gross
assumption that the object is infinitely self-similar. Consequently, fractal
dimension has been more often used to disprove hypotheses than to
conclusively prove them. Nevertheless, if two objects have similar fractal
dimensions over the same range of e values, and if one object is well
understood, one may use this information to help explain and analyze the
other object.
As an example, consider the dissipation of different pollutants in a
laboratory "atmosphere." In an isolated setting, the behavior can be studied
in great detail and the fractal dimension of the pollution clouds can be
determined over a wide range of e values. Observing the fractal dimension of
real-world pollution clouds over the same values of e might be a way to
determine which pollutant is controlling the clouds' behavior, and which
pollutants are escaping into the atmosphere. Similar techniques have been
used to liken the growth of certain plants to the formation of crystals in a
laboratory, and to extrapolate the behavior of the known system to that of an
unknown system.
72
Two other techniques for experimentally determining fractal dimension
are the area-perimeter relationship and the number-resolution relationship.
In the area-perimeter relationship, the area of a set of data is computed, and
its perimeter is measured several times on a finer and finer scale. In the
number-resolution relationship, the number of pixels on a screen is plotted
against the resolution of the screen for a variety of resolutions. In both of
these techniques, the results (graphs of ordered pairs of perimeters and scale,
or numbers and resolutions) are usually compared to those of a known set of
data points. If the slopes of these data points (or the logs of the data points)
correspond to an object with a known fractal dimension, then the fractal
dimension of the unknown object can be estimated. These two techniques
are used widely in practice, particularly with photographic images, but again
are most commonly used to determine the similarity between two fractal-like
objects.
Because the field of fractal geometry is so new, the possiblilties of
applications have only begun to be discovered; however, fractal dimension is
one aspect that has proven quite useful in many sciences.
J. THE KOCH SNOWFLAKE
An interesting phenomenon that has recently been explained using these
ideas is the discrepancy in the recorded lengths of the borders of many
countries. In Europe, the reported lengths of the borders between many
countries varies greatly, depending on who gives the report. For example, the
French records and the German records of the length of the French-German
border may differ by several kilometers. The reason for this difference is
73
simple. A border measured with a ruler from a 1:50,000 scale satellite
photograph records much less of the border's aetail than a surveying team on
the ground records, which again would record much less detail than a
government employee on the ground with a ruler. The smaller the device
used to measure the bc-.der, the more detail will be recorded, and the longer
the border will appear to be. This idea is readily illustrated through the
analysis of a fractal object called the Koch snowflake.
To construct a Koch snowflake, begin with an equilateral triangle, and
trisect each of its sides. At each point of trisection, construct another
equilateral triangle extending outwards from the original triangle, and trisect
each of their external sides (see Figure 3.28). Continuing this geometric
construction indefinitely yields the Koch snowflake. While the area inside
the Koch snowflake is clearly finite (since the entire snowflake lies inside any
circle containing the original triangle) the length of its border is, surprisingly,
infinite. To see this, note that at each step the length of the border is
increased by a factor of 4/3. Thus, starting with a length of L yields a final
length of limn_...L(4/3)nf, which is an infinite length. If we were to measure
the Koch snowflake with rulers of different scales, remarkably different
results for the length of the border will be obtained. This is the same
phenomenon being experienced measuring border lengths in Europe.
As another real-world application of this concept, many small "resort"
lakes entice tourists by advertising "a thousand miles of shoreline," or
promises to that effect. Never in such an advertisement will one find a
mention of the scale used to arrive at that number, as this is simply an
application of the above concepts.
74
Second StepRirst Step •
Start With
an EquilateralTriangle
Figure 3.28 Constructing the Koch snowflake.
K APPLICATIONS OF FRACTAL GEOMETRY
Aside from the applications already mentioned (namely, an aid to
computer graphics and the use of fractal dimension to classify and compare
similar objects) fractal geometry has so far found few other applications in the
physical sciences. As already mentioned, the infinitely-detailed property of
fractals does not hold in the real world when objects are analyzed at the
molecular and smaller levels. Even Benoit Mandelbrot, the German-born
American scientist who gave fractals their name, admits that true fractals do
not exist in nature. However, he is also quick to point out that there are no
truly straight lines or perfect circles either. While traditional geometries have
been inadequate in sufficiently describing all of nature, perhaps fractal
geometry, at least on the macroscopic level, may provide yet another
approach. When we consider the infinitude of natural objects that have a
fractal-like structure, we see that nature is indeed more fractal than it is
75
Euclidean. From the formation of crystals and snowflakes to the way coral,
certain roots, plants, and trees grow; from the shapes of lightning bolts and
some electrical discharges to cloud formations, weather patterns, and galactic
patterns; from the structure of our lungs and surfaces of our brains, to the
patterns of our veins; much of the physical world seems to have a fractal
structure to it. A more complete analysis of some of the minutely detailed,
self-similar objects in nature may someday be realized through fractal
geometry.
One area of mathematics, however, where fractal geometry has
unquestionably found a permanent role and distinguished itself is in the
study of chaotic dynamical systems. Dynamical (changing) systems are
prevalent in the real world and we are finding ever greater numbers of them
to be chaotic. In both mathematical models and the physical world, strange
sets of data points are being observed that seem to contain self-similarity and
infinite detail. Many of these sets of points are, in fact, fractals and with the
aid of fractal geometry we can understand them better. Hence, when we
realize how much of our universe is modeled by chaotic dynamical systems,
we will better appreciate the usefulness of fractal geometry in studying these
systems. We take up the study of chaos in the next chapter.
76
IV. CHAOS
A. INTRODUCTION
We have introduced and studied iterated function systems, from our
simplest example where fn+l(x) = (fn(x))2, to systems that produced very
intricate and interesting geometric shapes. The similarity between these
systems is that we could predict the exact behavior of the systems after any
number of iterations. Unfortunately, in nature this predictability seems to be
more the exception than the rule, which has caused scientists frustration for
centuries whenever their models of seemingly simple phenomena produced
erratic behavior. This observed phenomena in dynamical systems is called
chaos, a precise definition of which we provide below.
We introduce our study of chaotic dynamical systems by first looking at
discrete systems, such as the iterated function systems we have been
discussing. There are two reasons for this approach: the first is that discrete
systems provide a conceptually simpler setting for understanding the theory,
and the second is because many data are collected from the real world in
discrete increments. For example, consider a biologist studying the dynamics
of a population. Not only will the population change in discrete increments,
but the biologist can measure the population only at discrete intervals. In
fact, even in continuous real-world behaviors such as beam vibrations, we are
able to record measurements only at discrete intervals. Since much of science
involves studying a continuous real world based on discrete observations, it
77
makes sense to begin the study of chaotic dynamical systems with the discrete
case.
A further reason to study discrete iterated function systems is that they
capture the essence of an important real-world phenomenon known as
feedback. Feedback occurs when the present state of a system affects its future
state, which in turn means there could be at least a small time delay in the
change of the state of a system. For instance, feedback in speaker-microphone
systems causes jumps at discrete time intervals because of the time it takes the
sound to travel through the system (although we preceive these jumps as
continuous when we hear them). Biological systems which are periodic and
demonstrate feedback include predator-prey models, the motion of slime
molds, light emission by groups of fireflies, glycolysis and photosynthesis, and
even the conditions in the brain leading to epileptic seizures. Feedback is also
exhibited in electrical circuits, as the defense contractor TRW discovered
when chaotic feedback shut down its European computer network. Finally,
gaps in the asteroid belts that would be in phase with planetary orbits indicate
the effect of feedback in our solar system. These are just a few examples
where relatively simple mathematical models can capture important real-
world phenomena.
The following five sections are based on Devaney (1989), and provide a
background and a setting for studying chaotic dynamical systems. The
remainder of the chaper comes from a variety of sources, but in the cases
where the references differ on theorems or definitions, the ones used by
Devaney are followed. Additionally, many of the examples and applications,
78
*where not specifically referenced, come from Gleick (1988), Moon (1987), Levy
(1991), and Briggs (1989).
We begin by developing the mathematics of discrete dynamical systems.
Our goal is to understand thoroughly the idea of orbits of points under
repeated iteration.
B. GRAPHICAL ANALYSIS OF FIXED POINTS OF MAPS
We have already shown how graphical analysis of a function can reveal
information about its fixed points, periodic points, and whether those points
are attracting or repelling. We will now introduce the concept of graphical
analysis of the nth iterate of a function, denoted by fn(x). Consider first a very
simple function mapping the unit interval to itself f [0, 11-40, 11, defined by
fRx) = 2x (modl). The notation 2x (modl) means: multiply the input value
by two, and consider only the fractional part, i.e., the remainder upon
division by one. For example, M(1.57) = 3.14 (modl) = .14. This mapping gives
f(x)=2x if 0<x51/2 and f(x)=2x-1 if 1/2<x51. This function is called
the Baker map, and its graph is shown in Figure 4.1. The Baker map is clearly
an onto function, but it is not one-to-one since every y value in [0, 1] is the
image of exactly two values of x. Additionally, the fixed points of the Baker
map are 0 and 1, both of which are repelling.
Consider the second iterate of the Baker map, f2(x) = 4x (modi). Its
behavior can be seen more clearly by the graphical analysis shown in Figure
4.2. Since the interval [0, 1/21 is mapped to [0, 11 under the first iterate, it
makes sense that under the second iterate the interval [0, 1/2] will contain a
79
....f.x).,
0 1/2 1
Figure 4.1 The Baker map.
smaller copy of the entire first map. Additionally, the second iterate shows
the appearance of two new fixed points, 1/3 and 2/3. This fact demonstrates
the existence of a two-cycle in the original map; in fact, it is easy to verify that
f(1/3) =2/3 and f(2/3)=1/3. Notice also that the fixed points of f(x), 0 and
1, remain fixed for f2(x). Observing the fixed points of the nth iterate of a
map reveals both the periodic and fixed points of the original map. The third
iteration is f3(x) = 8x (modl) and its fixed points, 0, 1/7,2/7,..., 6/7,1 are
also shown in Figure 4.2. In the case of the Baker map, we can easily obtain a
dosed-form expression for its nth iterate: fn(x) = 2nx (modl). This formula
makes the Baker map very easy to graph and analyze, and we will return to it
when "chaos" is defined. It may be very difficult, however, to obtain a simple
expression for the nth iterate of a map, and it is frequently even more
difficult to graph it.
A slightly more complicated map to analyze graphically is the map
f(x) = 4x(1 - x), which is a parabola that maps the unit interval to itself. Its
80
graph and second iterate on the unit interval are shown in Figure 4.4. Again,
the appearance of fixed points under f2(x) shows a two-cycle of f(x), and a
simple check shows this two-cycle occurs at the points x = (5t *45)/8.
f(x) f(x)1...................... ......... 1.... ........ ... ... .... .
0 1/2 1 0x • £.x10 1/21
f 2 (x) - 4x(modl) f 3 (x) - 8x(modl)
Figure 4.2 The second and third iterates of the Baker map.
1 .. . . ...is a s .. ...... ...... .. .. . .. .. . .. .... .. .. .
i :
0 1/2 1 0 1/2 1
f(x) - 4x(1 -x) f 2(x)
Figure 4.3 The first and second iterates of fRx) = 4x(1 - x).
81
While this technique is not always easy to use (due to the difficulty of
graphing many functions accurately), it can reveal considerable information
about the orbits )f points in a dynamical system.
C. MAPS OF THE CIRCLE
Consider the map f:- SI-+S1 with f(9) = 29 of the unit circle to itself. Here
0 is the angle measured in radians (positive counterclockwise) between the
positive x-axis and the line joining the origin to any point on S1. Every point
on the circle is represented by the form 0 =- + 2k1 for any integer k. The
map f, which doubles the angle of any point on the unit circle, has certain
properties which help illustrate and suggest a precise definition of chaotic
systems (see Figure 4.4).
0=0
Figure 4.4 The map f(O) = 29 of the circle S1.
Fir-st notice that 0 = 0 is a fixed point of f, since f(0) = 0. Additionally, for
any point of the form 0 = 2kx/2n, fN(O) = 2kx, so that these points are
eventually fixed. The points of the form 9 = 2kx/2n form a dense subset of
82
S1, since between any two distinct points on S1, we can find a point of the
form 2kc/2n. To see this, consider two arbitrary distinct points 01 and 02 on
S1, where 01 = 2nal and 02 = 2=2 with 0 zq al < a2 < 1. Since a2 - al > 0,
there exists an integer m such that m(a2 - a,) > 1 by the Archimedian
property. Let 2n > m. Then 2n(a 2 - al) >1 implies there exists a positive
integer k such that 2ha 2 > k > 2hoal, or 02 > 2kx/2n > 01. Thus, points of the
form 2kx/2n are dense on S1.
The periodic points of f are slightly more difficult to find. Since
fn(0) = 2n0, the point 0 is periodic of period n if and only if 2n0 = 0 + 2k1
for some integer k. Solving for 0 yields 0(2n - 1) = 2k1, so for 0:< ks 2n - 1,
the points 0 = 2kx/(2n - 1) are the periodic points of period n. Using complex
analysis, it can be shown that these points are the (2n - 1) roots of unity, and
that they are dense on the circle S1. Geometrically, this means that the points
of period n are those that divide the unit circle into 2n - 1 equal segments.
This interpretation also shows that they are dense on SI. This is shown
graphically in Figure 4.5 for the points of period two. The point O0 trivially
has period two since fn(O0) = 00 for all n. The points 01 and 02 have period
two since f(RO) = 02 and f(02) = 01.
Another interesting property of f is that for any two points on S1, no
matter how close together they start, iteration under f will separate their
iterates by an arbitrary amount The formal definition of this characteristic is
as follows: a mapping f: S-*S has sensitive dependence on initial conditions
if there exists 8 > 0 such that for any point x e S and any neighborhood N
83
". 0 -2n/3
i e2 = 4x/3
Figure 4.5 Periodic points on S1 of period two.
of x, there exists a point y e N and an integer n > 0 such that
d(fn(x), fn(y)) > S. Notice that under this definition, it is not necessary for
every point in the neighborhood N to move away from x under repeated
iteration. However, in any neighborhood containing x there must exist at
least one point that does move away from it Since our map of the circle
f(0) = 20 doubles the arc, length between any two points, there is always an
iteration number n that moves two points at least a given distance 8 apart;
hence the map f(0) = 20 has sensitive dependence on initial conditions.
A third interesting property of the map f(0) = 20 is that for any pair of
open arcs U, V c S1, there exists a number n > 0 such that the intersection of
fn(U) and V is nonempty. The formal definition of this property is as
follows: a mapping f: S--S is called topologically transitive if for any pair of
open sets U, V c S there exists an integer n > 0 such that fn(U) n V * 0.
Since any arc U in SI is expanded under iteration of f(0) = 20 until it
84
covers all of S1, every arc eventually intersects every other arc in S1. Thus
the mapping f(O) = 20 is topologically transitive.
The three properties of f: S1¼S1 where f(O) = 20 define a dynamical
system to be chaotic. The formal definition follows.
D. CHAOTIC DYNAMICAL SYSTEMS
A mapping f: S-4S is called chaotic if the following three conditions are
met:
1. f has sensitive dependence on initial conditions;
2. f is topologically transitive; and
3. Periodic points are dense in S.
While precise definitions of chaos vary slightly among different
disciplines, most of them are at least similar to those we gave here. These
three conditions are frequently referred to as unpredictability,
indecomposability, and an element of regularity, respectively. Because of the
sensitive dependence on initial conditions, it is almost impossible to predict
the orbit of an arbitrary point with any degree of accuracy. This is the
unpredictability aspect of chaos, and it makes numerical computations with
chaotic dynamical systems virtually impossible (since the slightest rounding
error at any step will almost certainly create a point with an entirely different
orbit). The indecomposability property comes from the fact that there is no
way to decompose the set S into two disjoint subsets that do not eventually
interact (since any open subset of S eventually intersects every other open
subset of S under iteration of f). Finally, the denseness of the periodic points
provides an element of regularity amidst all this chaotic behavior.
85
Let us now demonstrate that the Baker map is chaotic directly through a
graphical analysis. To see that the periodic points are dense, note that the nth
iterate fn(x) = 2nx (modi) has n evenly spaced fixed points in [0, 11 for n
arbitrarily large as indicated in Figure 4.2. Figure 4.6 also shows that f is
topologically transitive since any open interval (p, q) contains an interval
that is mapped to [0, 11. Finally, Figure 4.7 shows the sensitive dependence
on initial conditions, because any two points p and q are moved an arbitrary
distance apart under some iteration of f. While this graphical technique is
not a proof, it can be supported by a rigorous mathematical argument. It
alsodemonstrates; the usefulness of analyzing some maps graphically to
determine if they are chaotic.
p q x (not to scale)
Figure 4.6 Topological transitivity of the Baker map.
86
"f(x)
f(q)
f(P)
0 -
p q x (not to scale)
Figure 4.7 Sensitive dependence on initial conditions of the Baker map.
Chaos occurs not only in mathematical equations, but in fact seems to be
most prevalent in the real world. While accurately modeling real-world
phenomena with mathematical equations is sometimes very difficult,
particularly when the physical system is chaotic, it happens that even a
simple model crudely representing a physical system may turn out to be
chaotic. This happenstance in turn tells us that the more complex physical
system is probably also chaotic. We will see this happen when we discuss the
Lorenz equations as a crude model for weather prediction. Another example
of this chaotic behavior occurs with a simple differential equations model
used to describe beam intersections in particle accelerators. The model
explains why it is so difficult to predict the action of intersecting beams in
particle accelerators (at least using existing models).
There are several examples of physical systems where chaos has been
observed,,studied, and in some cases, somewhat understood. These systems
include turbulence in fluids, thermal convection in gasses, panel flutter on
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supersonic aircraft, certain chemical reactions (specifically the Belousof-
Zhabotinsky reaction), abnormal cardiac rhythms, nonlinear electrical
circuits, biological population dynamics, vibrations of buckled elastic systems
(such as beams), geomagnetic field reversals, and even planetary motion.
Many of these systems are discussed in Moon (1987), Holden (1986), and
Rasband (1990).
Unfortunately, it is very difficult to identify chaotic systems
experimentally. One reason is that numerical roundoff in collected data
could lead to an erroneous assumption that two data points are the same
(indicating a cycle), whereas a slight difference in their actual values may
cause their orbits to diverge rapidly (because of sensitive dependence on
initial conditions). An even more difficult problem is with distinguishing
between a truly chaotic orbit and a cycle with a very long period. To
emphasize the significance of this problem we note that the Department of
Mechanics at Cornell University requires 4,000 non-cyclic real-world data
points before scientists there declare a system as being chaotic. For biologists
and economists, who model annual trends, this requirement may seem
unreasonable. However, to a helicopter pilot whose life depends on chaotic
vibrations not occurring on the main rotor blade, this requirement may not
be stringent enough.
While the map f: S1-+S1 with f0O) = 20 provides a simple example of a
chaotic map, it by no means represents the extent of chaotic dynamical
systems. We turn now to several more interesting chaotic maps.
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E. TOPOLOGICAL CONJUGACY
One way to determine if a map is chaotic is to check directly the three
conditions of chaos. However, using the concept of topological conjugacy, we
can apply the dynamics of a familiar map to those of another function. Two
maps, f: X-4X and g: S--S are said to be topologically conjugaie if there exists
a homeomorphism h: S---X such that f(h) = h(g). If f and g are
topologically conjugate, then g = h-lfh, so gn = (h-lfh)n = (h-lfh)(h-lfh)...
(h-lfh) = (h-1f)(hh-1)f(hh-1) ... f(hh-1)(fh). Since h is a homeomorphism,
hh-1 is the identity function, hence gn = h-lfnh, and hgn = fnh. It follows that
f and g share the same dynamical properties; in particular if g is chaotic the
same is true of f. This idea is shown in Figure 4.8 with. If h: S--X is not a
homeomorphism (e.g., if it is two-to-one) then f and g are said to be
topologically semi-conjugate. Nevertheless, if g is chaotic, f is still also
chaotic. Since h is two-to-one, the dynamics of f are even more complicated
because h introduces even more periodic points or cycles. It suffices that
chaotic behavior is preserved through topological semi-conjugacy, which is
the extent to which we use this concept.
S - g--+ S
h .h
X f -- X
Figure 4.8 Diagram showing topological conjugacy between g and f.
What follows is an example of the usefulness of topological conjugacy.
Consider the map f: R-+R, where fRx) = 2x2 - 1. We would like to know if
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this map is chaotic. We could try to show sensitive dependence on initial
conditions, topological transitivity, and search for a dense orbit. Instead we
consider the mapping h(O) = cosO. Since, cos20 = 2cos20 - 1 = f(h(O)), the map
f is topologically semi-conjugate (since the cosine function is not one-to-one)
with the map g(0) = 20 when g: S1-S1, as shown in Figure 4.9. Since we
know that g is a chaotic map from our previous discussions, and that f
shares the same dynamical properties as g, the map f(x) = 2x2 - 1 is also
chaotic.
0 - g -- 20
1,h Ih
cosO - f -- cos2O
Figure 4.9 Topological conjugacy between g(e) = 20 and fRx) = 2x2 - 1,under h(0) = cosO.
We frequently use the concept of topological conjugacy to analyze maps
suspected to be chaotic. The simplicity of the previous example shows why
topological conjugacy is often used.
F. CHAOTIC DYNAMICS ON CODE SPACE
Consider now the shift map a: 12-+12 on code space 12 = (sls2s3...:
Si e (0, 1)) where a(sls2S3...) = s2s3s4.... We show directly from the
definition that this mapping is chaotic.
To see that 0: ,7,2-+2 has sensitive dependence on initial conditions,
consider any point x = xlx2x3... in £2. For a point y = yly2y3... to be in a
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neighborhood of x, it must agree with x for a certain number of digits
depending on the size of the neighborhood. Suppose the agreement is with
thefirst n digits; x =yl, x2=y2,....,xn=Yn. Ifwechooseyk* xk forall
k > n, then y still lies in the neighborhood of x, but I on(x) - an(y) I > 8 for all
8 < 1. Hence, the point y, which is arbitrarily close to x, moves arbitrarily
away from it under iteration of c. This establishes sensitive dependence on
initial conditions.
To show topological transitivity, consider two open sets U, V e 7,2. Since
U is open, its points U1U2U3...Un... in 7,2 agree for the first n digits, but
can differ in any way beyond the nth digit for some positive integer n.
Likewise, points in -2 which agree in the first m digits for some m all lie in
V. It follows that the point U1U2U3. ...UnVlV2V3. .. Vm... lies in U, while the
nth iterate of U under a contains the point vjv2v3.. .vm... in V. Thus the
intersection of on(U) and V is nonempty for any open sets U and V of 12.
Therefore, o: 12-+12 is topologically transitive.
To see that the periodic points of a are dense in 12, we show that for any
point s e 12 there is a periodic point arbitrarily dose to it. Thus, let
s = sjs2s3.... We want to find a periodic point x that corresponds to s for
up to n digits (selecting n arbitrarily large makes the periodic point
arbitrarily close to s). Choose the point x = S1s2. . SnSIS2 ... Sn. . , which is a
periodic point of period n. The point x is arbitrarily dose to S. Hence, the
periodic points of a are dense in 12.
In conclusion, the shift map on code space a: 12--+12 is chaotic. Like the
map of the unit circle f(8) = 20, the shift map is quite useful in showing that
other maps are chaotic through the use of topological conjugacy. When we
91
I!
can establish topological conjugacy between the maps f and a, we will refer
to analyzing the behavior of points under f as studying the symbolic
dynamics of the map f.
Using the concept of topological conjugacy, it is now an easy exercise to
show that the Baker map f(x) = 2x (modl) is chaotic. This behavior can be
shown through either the shift map on code space or the angle doubling map
on the unit circle, which confirms our earlier geometric demonstration of
this fact.
G. NEWTON'S METHOD FOR X2 - -1
To see how chaos sometimes arises from mathematical systems, consider
the following example from numerical analysis due to Strang (1991). Recall
that in finding the zeros of a real valued function f(x) by Newton's method,
an initial guess together with the iterative process Xn+1 = Xn- f(xn)/f(xn) is
used. Applying this method to find the roots of f(x) x2 + 1, the iteration
equation reduces to the iterated function system Xn+l = Xn - (xn2 + 1)/(2xn) =
(1 /2)(xn - /xn). Newton's method converges for most polynomials under
certain conditions (e.g., when F(x) * 0 and no inflection point occurs between
the root and the initial guess). In the case in question, the only roots are ±i
so the system cannot converge on the real line. (But note that f(x) is always
strictly positive and has no real roots.)
The behavior of the iterated function system changes dramatically
depending on the initial guess. For some values of xo, this system diverges to
infinity; for example, if x0 = 1, then xl = 0, and x2-+* (since we cannot
divide by zero). On the other hand, if we choose xo to be very large, then xi
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"is approximately half that value, and the orbit moves towards zero until the
1 /Xn term makes the next iteration large again. Finally, if we choose
xo = 1/,.r, then x1 = -1/43 and x2 = 1/r3-, yielding a two cycle.
Because we are searching for a root value which is nonexistent on the
real line, Newton's method exhibits strange behavior for x2 + 1 = 0. We now
explore whether this system is chaotic. From trigonometry, cot2O =
(1/2)(cote - 1/cotO). Using the map h(O) m cotO it follows that the map
g(0) = 20 is topologically conjugate to f(x) = (1/2)(x - 1/x), as shown in Figure
4.10.0 - g-- 201 h Ih
cotO - f -4 cot28
Figure 4.10 Topological conjugacy between g(e) = 20 and f(x) = (1/2)(x - 1/x)under h(e) = cotO.
Thus fRx) exhibits the same dynamical properties as g(O) = 20 on SI; hence f
is chaotic. We can actually observe this behavior graphically by considering
the graph of the iterates of f. Letting xo = cote, we obtain xI = f(xo) = cot2Q,
x2 = cot4O, and in general, xn = cot(2ni). Hence, the orbit of x under f is the
sequence cotO, cot2e, cot40... for different values of 0 corresponding to our
initial guess. The graph of cotO is shown in Figure 4.11, and helps to
demonstrate the results noted earlier for different initial guesses.
If we start with xo = 1 we get Oo = x/4 (since cot (x/4) = 1). Then 01 = x/2,
corresponding to xi = 0 (since cot (x/2) = 0). Next 02 = z and the orbit for x
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f (0)
Figure 4.11 The cotangent map and a chaotic orbit.
diverges since cot x is undefined. In fact, any starting value of xo
corresponding to the form O0 = ks/2n, for positive integers k and n, results
in an orbit that eventually diverges to -.. Starting with xo = 1 /F, we find
Oo = x/3 (since cot(x/3) - 1/13). Then 1 -200= 2X/3 gives xi -1 i, (since
cot(2x/3) = -11V). Next 02 = 20 1 = 4U/3 = o + x. Since the cotangent function
is periodic with period x, x2 = I/i/3. Therefore, this orbit is the two-cycle we
observed earlier. In fact, if Or = (p/q)x, where p/q is not of the form k/2n,
then the orbit eventually cycles. This observation further demonstrates the
denseness property of the periodic points. Finally, if 00 is an irrational
multiple of x, the orbit will never diverge or cycle; in fact, it will be chaotic, as
we already knew.
Considering our search for i on the real line, the points that we hope
would eventually converge to zero are yn = Xn2 + 1. If we further analyze
these points, we find yet another chaotic map which shares the dynamics of
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"the one we just studied. For Yn+l = (Xn+1) 2 + 1 =(1 /4)(Xn -1 /xn) 2 +1 =
(I/4)(xn2 + 2 + I/xn2). Simplifying algebraically, Yn+1 = (1/4)(xn 2 + l)2 /xn 2 =
yn2/4(yn - 1). If we change variables and let z = l/y, this last equation reduces
to zn+l = 4zn(l - zn). The latter iterated function is a member of the quadratic
family of maps we study in great detail below, and we know it is chaotic.
As a final illustration of Newton's method, we analyze the system
f(z) = z4 - 1 in the complex plane. Since this equation has four roots, ±1 and
±i, depending on the initial value, the iterations could end up at any one of
the four roots. The results encountered with computers when searching for
these roots has not always been predictable, depending on the initial value.
Now each of these roots has around it a 'basin of attraction" inside of which
all initial values converge to that root upon repeated iteration. However,
along the diagonals Re(z) = Im(z) and Re(z) = -Im(z), there are regions that
give rise to chaotic behavior. To see this, color the complex plane based on
which initial values converge to which root: that is color all the points that
converge to 1 blue, to -1 green, to i yellow, and to -i red. There are four
large basins of attraction with infinitely detailed, multicolored borders
between them. In fact, there is no dearly defined boundary between any two
basins of attraction: between any blue and red region, there is a yellow and a
green one. A schematic of the complex plane showing the basins of attraction
and the chaotic regions is shown in Figure 4.12. A detailed color image of this
figure can be found in Gleick (1988, p. 114).
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Yellow Im(z) Chaotic
(Shaded)\Yellow
jRegions
S1 i Re(z)
Gree Blue
• "Red ,
Figure 4.12 Schematic of the complex plane indicating the behaviorunder Newton's method for z4 + 1 and the (colored) basins of attraction.
H. THE QUADRATIC FAMILY OF MAPS
In this section we study a specific case of maps from the quadratic family
f(x) = ax2 + bx + c. We discuss a model from population biology to develop
this map, which comes from Briggs (1989). The mathematics in this section is
from the book by Devaney (1989) and the article by Devaney (1989), while
some of the more detailed results, particularly about the Feigenbaum
constant, are from Rasband (1990).
Consider a population of moths which live for one season, lay their eggs,
and then die, all in a fixed and limited environment. If we do not consider
the interaction of the moths with the environment or other species, we can
model the dynamics of the moth population by assuming that the size of the
population increases after each life cycle in constant proportion to the size of
the population during the previous cycle. If we let the constant of
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proportionality equal B (for birthrate), then for each population cycle we
have xn+i = Bxn, where x represents the population and n represents the
time period. Given only these conditions together with B > 1, this model
predicts the population increases forever and eventually becomes arbitrarily
large. Clearly no population in the real world is represented by this model,
since no limited environment can sustain an infinite population. Thus it is
necessary to adjust the model.
Suppose then that as a population increases above the maximum
sustainable capacity of the environment, the competition for resources causes
some of the population to die off or be killed. Hence a particularly large
population might cause the population to actually decrease in the next cycle.
If we scale our model so that 0 < x < 1, with 1 being the capacity of the
environment, then incorporating the multiplicative factor (1 - x) into our
model could account for the limitations of the environment on large
populations. We now adjust B to account for the birthrate (and deathrate) in
a fixed environment and denote its new value by X, obtaining the refined
model xn+l = ;xn(l - xn). This model shows that for small populations, the
population growth is dose to the original model xn+l = Bxn. However, as the
population approaches the maximum sustainable capacity of the
environment, its growth starts to decline because of the (0 - xn) factor.
The equation Xn+l = )Xn(1 - xn), X > O, is called the logistic equation. It is
perhaps one of the simplest population models to represent the interaction
between a species and its fixed environment (although it excludes
interactions with other species). Allowing x to vary between zero and one
scales the model to any population size. Restricting the parameter 0 : X <4,
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we see below that the model never allows the population to increase above
the maximum capacity of the environment. The logistic equation has been
used by biologists for many years, and it has been a major goal to understand
populations that are modelled by this equation. One methodology has been
to determine experimentally a value of X based on the species and the
environment, and then to observe the population over time to see if it
behaves according to the model. This simple model is a fairly good predictor
for a simple species like bacteria or yeast growing in a culture, but it has not
always correctly predicted the observed results for more highly developed
species, like mammals. Some populations have been observed to settle down
to a fixed value for all time increments, others have been seen to cycle
between two, four, or even larger periods, and still others have demonstrated
completely unpredictable behavior that fit no known model at all. What
happens is that, for certain values of X, the logistic equation f(xn) = Xn( - Xn)
becomes chaotic. Thus it is impossible to predict its behavior even after a few
iterations. We next examine this map mathematically.
First note that for values of xo equal to 0 or 1, the next iteration xI = 0
is a fixed point of this map, since f(O) = 0. Also, for values of xo outside the
interval [0, 1], xl becomes negative (which does not make physical sense
since we are studying populations). Furthermore, all negative values diverge
to --- under repeated iteration of E Hence, for practical reasons we
concentrate on the dynamics of this system on the interval [0, 1].
Suppose 0 <X < 1. Then x = 0 is the only fixed point of the system (see
Figure 4.13). Moreover, all values of x0 e [0, 11 tend to 0 under iteration of
f. (Recall that if I f(x) I < 1 for a fixed point x, then the fixed point is
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attracting. Since f(x)= (1 -2x), for X < 1, we obtain F(0) = X < 1; so 0 is an
attracting fixed point). The physical interpretation of this result is that the
population does not reproduce fast enough to sustain itself and eventually
dies out.
f(x) f(x)
1 ............................. fx x
0 1
Figure 4.13 The logistic equation for X < 1.
Now increase X so that it is greater than 1. Then 0 becomes a repelling
fixed point since F(O) = X > 1. However, note also the introduction of another
fixcd point p in the interval [0,1], given by p = 1 - 1/A (see Figure 4.14). If
we consider values of X such that 1 < X < 3, we see that the point p is an
attracting fixed point since If'(p)I = IM(1-2+2/A)I = 12-XI <1. Hence, for
values of X between 1 and 3, 0 is a repelling fixed point and p = 1 - 1/A is
an attracting fixed point. This means that our population will always settle
down to a fixed value and not vary in cycles.
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f (x) 1. . f(x) x
0 X 0
Figure 4.14 The logistic equation for I < X < 3.
Now let X jump to the value X = 4. Since xn+l - 4xn(l - xn) we know
that this map is chaotic because of our discussion of Newton's method for
Z2 . -1. We now demonstrate this idea graphically to clarify further what is
actually happening. Consider the graph of f(x) - 4x(I - x) on the interval [0,
11 in Figure 4.15. The graph represents a two-to-one and onto function.
Hence, the second iterate f2(x) creates a condensed copy of f(x) on each
interval [0, 1/21 and [1/2, 11. Moreover, that f2(x) has four fixed points (see
Figure 4.16). The third iteration fO(x) is also shown in Figure 4.16. Observe
that we are creating a graphical situation similar to the Baker map studied
earlier (with parabolic arcs rather than straight lines). The graphs reveal that
f(x) has sensitive dependence on initial conditions, topological transitivity,
and a dense orbit; so it is, in fact, chaotic. This analysis explains why, for some
populations, the observed values fluctuate wildly, despite a seemingly simple
mathematical model describing the behavior.
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f(x) f (X) -x1 .............. ".. . . . .. f x
0o x0 3/414
Figure 4.15 The logistic equation for X =4.
f(x) f 3(x)..................... ....... ......... 1.. ....... ........
S---'--- - x •.- x0 0
Figure 4.16 The second and third iterates of the logistic equation for X =4.
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* Another way to show that the map f(x) = 4x(1 - x) is chaotic is to use the
concept of topological conjugacy. Recalling the trigonometric identity
sin220 = 4sin2O(1 - sin 20), the mapping h(0) = sin 20 shows that fRx) = 4x(1 - x)
is topologically semi-conjugate with the map of the circle g(0) = 20. This
commutative diagram is shown in Figure 4.17. This is yet another way to
show that the map fRx) = 4x(1 - x) is chaotic.
0 - g -4 20
.Lh 4,h
sinO -2 f - sin20
Figure 4.17 Topological conjugacy between g(O) = 20 and fRx) = 4x(1 - x).
Now examine what happens when 3 < X < 4. We observed an attracting
fixed point for , < 3, and chaotic behavior for X = 4. Thus as X increases
from 3 to 4 the dynamics of the systeni must change dramatically to
produce this chaotic behavior. For X = 3, the magnitude of the derivative at
the fixed point p=1-1/7,=2/3 isequalto 1fF(2/3)1 = 12-7XI =1; for X >3,
I f(p) I > 1. This means that as the value of I passes through 3, the fixed
point p goes from attracting to repelling. This result implies that both of the
fixed points, 0 and p = 1 - 1/A are repelling. With both fixed points
repelling, one might think the orbit will fluctuate wildly within the interval
[0, 11. But what really occurs is the creation of an attractive two-cycle, as
shown in Figure 4.18. This splitting of a fixed point into an attractive cycle is
called a period-doubling bifurcation.
102
f(x)
0 10
Figure 4.18 An attractive two-cycle in the logistic equation for X > 3.
The two-cycle previously described is one of the phenomena that puzzled
biologists for many years. Physically, a small population would flourish and
breed very rapidly, since the environment could easily support it. After a
time a large population would be created in the next cycle. This population
would breed more sluggishly because it was nearer the capacity of the
environment. Furthermore, the population would not approach its fixed
value (since that value was repelling). Even when biologists altered the
population it would still tend back to the attractive cycle.
As we continue to increase the value of I beyond I = 3, we find that at
S- 3.4495 the two-cycle becomes repelling and an attractive four-cycle is
created. Now, as with X = 3, the derivative I (fn)'(p) I for the cyclic points Pi
and p2 is passing through 1, and the two-cycle becomes repelling; another
bifurcation of the system has occured.
If we continue to increase X., the next bifurcation occurs at X - 3.544,
where the four-cycle becomes repelling and an attractive eight-cycle is created.
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Plotting the values of the parameter X against the values of the attractor (the
attracting periodic points of period two, four, eight,...) shows how each cycle
bifurcates into another cycle with twice the period as the previous one. Such
a bifurcation diagram is shown in Figure 4.19 for 2.9 < X < 3.9. The parameter
values at which the next four bifurcations occur are X - 3.5644, 3.5688, 3.5697,
and 3.5699. As one can see, these numbers are getting closer together and they
converge to a value where the map becomes chaotic.
IS
Figure 4.19 The bifurcation diagram for the logistic equation as Xincreases. The figure is from Holden (1986, p. 46).
The apparent convergence of these numbers led to the discovery of a
constant that appears to be almost universal in dynamical systems. Denote
the values of X at which bifurcations occur by Xk for k = 1, 2, 3,..., denote
the value to which they converge by A., and consider the following. Figure
4.20 shows the second iterate of the logistic equation. Note that it contains a
scaled copy of the first iterate, and that each further iterate will again contain a
scaled copy of the original map. This geometric observation provides insight
to the phenomenon of period-doubling. At X = Xk-1, a period-doubling
occurs in the system that lasts for a duration (in X) of (Xk - Xk-1). The next
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period doubling occurs at X = Xk, and lasts for a ,-duration of ()Lk+1 - Xk). It is
not unreasonable to expect a "scaling" in the successive .-lapses based on the
graphical observation. Hence, if we assume the convergence of (. - X•.) to
be geometric, then ;.. - .k = c/Sk where c is a constant and 8 is a constant
greater than one. Using this to solve for 8 in terms of the .k, we see that
8 = (.k -. k-l)/(;.k+1- -;k). From our previously computed values of Xjk, we
see that 8 - 4.6692, which is called the Feigenbaum constant, named after the
American scientist and mathematician Mitchell Feigenbaum, who discovered
it. Using this constant to solve for X., we find that X. - 3.5699456. Hence,
with X = A.., the logistic map is chaotic. (As an aside, the Feigenbaum
constant has been found in many different dynamical systems. For example,
models for electrical circuitry, optical systems, and economic cycles, as well as
physical systems such as chemical reactions, erratic heart behavior, and even
dripping faucets, have all exhibited the Feigenbaum constant when viewed in
the proper phase space. With its value now known, the number of
dynamical systems where the Feigenbaum constant is found is increasing
quite rapidly.)
- 2f(x f_ • (x)
0 X 10 X
Figure 4.20 The first and second iterates of the logistic equation.
105
I'
This "period-doubling route to chaos" was one of the first routes to chaos
understood by mathematicians. It has helped biologists understand the many
different results associated with systems modeled with the logistic map.
While this discussion about bifurcations and the period-doubling route to
chaos has been mostly geometric and intuitive, there is a rigorous underlying
mathematical theory validating these concepts. However, this theory
requires an understanding of "kneading theory," an elaborate version of
symbolic dynamics that is beyond the scope of this thesis.
While the logistic map is very restrictive (in that it models only a single
species interacting with a fixed environment) it still provides a great deal of
information about population dynamics. When the model is further refined
with the introduction of another species (i.e., a predator-prey or plant-
herbivore model), the possibility for chaotic behavior increases with the
complexity of the model. Also, the logistic equation has been used by medical
scientists to model the spread of infectious diseases (the x term being the
number who are contagious and the (I - x) term being the number who have
developed immunity), and even by sociologists to model the spread of
rumors.
We now continue our mathematical analysis of the logistic map beyond
the ranges of X that correspond to physical population systems. Specifically,
we examine what occurs with the map Xn+l = ,xn(l - Xn) when X increases
above 4. The simplest way to observe the dynamics is graphically, as shown
in Figure 4.21. As before, all initial values outside of [0, 11 diverge to -m
under infinite iteration. Notice now, however, that there is a set of points
inside [0, 1] mapped outside the interval after the first iteration; thus they
106
"diverge to --. Examining the second iteration, we see that the two
subintervals remaining after the first iteration are mapped to [0, 1]; each will
then contain a smaller copy of the original map, and hence a subinterval
mapped outside of [0, 11 (see Figure 4.22). In fact, if we consider the set of
points which do not eventually diverge to -.o, by this construction we obtain
a Cantor set of points remaining in the interval after infinite iteration for
X > 4. Values of the interval [0, 1] mapped outside of the unit interval, and
hence diverging to -@, are said to be values that escape under iteration of f.
The value of X for which the interval (1/3, 2/3) escapes under the first
iteration of f can be found by letting 1 = W(1/3)(1 - 1/3), which yields X = 9/2.
However, iterating the logistic equation with X = 9/2 will not yield the
classical Cantor set, since the subintervals created in the second iteration will
be skewed and will not be exact middle-thirds intervals. In fact, the point
f(X) Af(x) - x
Figure 4.21 The logistic equation with X > 4 and the interval that escapes.
107
f(x) f(x) - x
0Figure 4.22 The second iterate of the logistic equation with X > 4.
x = 1/9 escapes under the second iteration. However, we can also see how a
Cantor-like set of points remains in the set. Considering X = 9/2, we see that
f(1/3) = 1, and since f(1) = 0, with 0 being a fixed point of the map, the point
1/3 forever remains fixed in the set. For X > 4, the logistic equation is chaotic
on a Cantor subset of [0, 11, while on the complement of this set, all orbits
diverge predictably. This contrasts with the logistic equation with 1 = 4,
which is chaotic on the entire interval [0, 11.
L DIFURCATIONS
In the last section we introduced the concept of bifurcations as period-
doubling phenomenon leading to chaos in the logistic map. We now study
the general concept of bifurcation and its interpretation in the real world.
The mathematical theory in this chapter is from Rasband (1990),
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"Guckenheimer (1990), and Seydel (1988), while the particular definitions and
theorems are from Devaney (1989). The examples are taken from Moon
(1987) and Holden (1986).
Bifurcations are normally associated with a change in the physical state of
a system. In population dynamics, one bifurcation is exhibited by a
population changing from tending towards an attracting fixed point to
tending towards an attractive cycle. Changes in the states of other physical
systems (such as water cooling until it freezes, a beam bending under a load
until it buckles, a balloon being blown up until it bursts, or even the crash of
the stock market) are also examples of bifurcations. Clearly, some of these
behaviors have such complex and intricate mathematical models that it
would be virtually impossible to quantify them and determine the exact
nature of the bifurcation. Nevertheless, there are several systems which have
been modeled successfully and for which the mathematical analysis of
bifurcation accurately predicts gross changes in the state of the system.
Mathematically, there are several types of bifurcations. FMist we discuss
those that occur in one dimension (as with our previous example of the
logistic equation). In the logistic equation, the bifurcation occurs as the
parameter value X is varied until it passes through certain values which
changed dramatically the behavior of the system. This example suggests we
consider families of functions of real variables, where the functions depend
"smoothly" on a parameter. Thus, define g(x, ).) = fj(x), where f(x) is a C-
function of x for each fixed X., and g(x, X) depends smoothly on L The
logistic equation g(x, X) = f;(x) = Xx(1 - x) is one such family.
109
We first present a general result about bifurcation theory that applies to
all one-dimensional bifurcations. Then we give specific examples of typical
bifurcations and physical systems that behave similarly in the real world. The
general result is that bifurcations occur only near non-hyperbolic fixed and
periodic points. (Recall that a hyperbolic periodic point is one where
I (fn)'(p) I * 1, where n is the prime period of the point, for a fixed point,
n = 1.) While the theory applies to both fixed and periodic points, we present
it here only for fixed points. Recalling further that a fixed point p is an
attractor if I f(p) I < 1, and a repellor if I f(p) I > 1, we would expect to find
interesting behavior as I f(p) I passes through 1. Such points p are the only
points where bifurcations can and do occur.
BIFURCATION THEOREM. Let f). be a one parameter family of functions,
with p a hyperbolic fixed point for some fixed X& i.e., fo(p) = p and
f',.(p) * 1. Then there exist intervals, I about p and N about I, and a
smooth function h: N--I such that h0.0) = p and f;(h(.)) = h(X). Moreover,
fX has no other fixed points in L
PROOF. Consider the function defined by G(x, X) = fx(x) - x. By hypothesis,
G(p, )4 = 0, and oDG/Nx I (p, )o) = f•'(p) - I 0. By the Implicit Function
Theorem, there are intervals, I about p and N about Xo, and a smooth
function h: N--I such that ho) = p, and G(h(;.), X) = 0 for all X E N.
Moreover, g(x, X) * 0 unless x = h(X). This completes the proof.
110
A graphical demonstration of this theorem is shown in Figure 4.23. The
graph shows that nearby graphs fxj and fX must have the same property for
a sufficiently small interval about AO, because the graph of fX meets the line
y = x at an angle at (p, p) and since f varies smoothly with X. Hence, there is
only one fixed point near p for X in some neighborhood of ko. At this
point there is a function g;. that is topologically conjugate to fx via the map
H.(x) = x - h(X) for which the origin is always a fixed point. This fact allows
us to study maps with 0 as a fixed point, and apply the results to any map fL
with a nonzero fixed point. (To see this, consider fJ with f(h(X)) = f(X). If
g)L(x) = f)(x + h(O)) - h(X), then gX(O) = f)(h(,)) - h(;) = 0 for all X, so 0 is
always a fixed point for g.) Thus it suffices to present our complete results for
one-dimensional bifurcations using functions whose fixed point is the
origin.
- 2S~f
Pl Ip P2
Figure 4.23 Schematic of the Bifurcation theorem.
111
Now that we know that bifurcations occur at non-hyperbolic periodic
points, we are prepared to study specific types of one dimensional
bifurcations. The first, and perhaps mathematically the simplest type, is the
tangent bifurcation (also called the saddle-node). The family fI.(x) = Xex, for
X>O, has a bifurcation at x=O when X=I/e. If X> 1/e the function f• has
no fixed points, and f;Ln(x) -- for all x. When IL = 1/e, the function has one
fixed point at p = 1. This point is attracting from the left and repelling from
the right. This last condition, which may occur for a non-hyperbolic fixed
point, is called semi-stable. For values of X < l/e, f). has two fixed points, pi
and p2, with pi attracting and P2 repelling. Graphs of these three cases are
displayed in Figure 4.24, and the graphical analyses are left to the reader.
fi(X) f (X) f (X)
xt- xP1 P2
Figure 4.24 Graphs of f(x) =Iex with (1) X < 1/e (2) X = 1/e (3) X)> 1/e.
We can see that this bifurcation occurs as the graph of fx becomes
tangent to the line y = x. First one, then two, fixed points emerge as the
parameter passes through the critical value. We can also verify that this
112
bifurcation occurrs at a non-hyperbolic fixed point p = 1, since fi/e'(l) -
(1/e)el = 1. This bifurcation gets its name from the way f) approaches the
line y = x tangentially. The bifurcation diagram for the function fi/e,
plotting the fixed points p on the vertical axis versus X is shown in Figure
4.25.
x
11
1/e "(not to scale)
Figure 4.25 Bifurcation diagram for f(x) = Xex.
The mathematical conditions guaranteeing the occurrence of a tangent
bifurcation, and the resulting bifurcation, are as follows.
THE TANGENT BIFURCATION. Given that
1. (o) = o,
2. 4o(0) = 1,
3. f'o"(0) *0, and
4. ok/A I.•o •10.
113
'Then there exists an interval I about 0, and a smooth function h: I-+R
satisfying h(O)= XO, such that fh(x)(X) = x. Moreover, h'(O) = 0 and h"(O) *0.
The tangent bifurcation has received considerable attention of engineers
recently because it is known that a beam or small arch bending under a load
undergoes a tangent bifurcation at its buckling point Additionally,
astronomers have identified the equilibrium response of massive cold stars
with the tangent bifurcation.
A bifurcation familiar to us from the analysis of the logistic equation is
the period-doubling bifurcation. We now give another example of the
period-doubling bifurcation. Consider the family of functions f.(x) Xex, for
X < 0. Graphs of these functions are shown in Figure 4.26. When X = -e,
fL(-I) = -1, and I f'(-I)I f I-e(e-1)I = - I- i 1; thus p = -1 is a non-hyperbolic
fixed point. We would expect a bifurcation to occur as X passes through the
critical value -e, and indeed one does occur. When X > -e, the fixed point p
is attracting, and when X < -e, it is repelling. Hence, the nature of the fixed
point changes as X passes through -e, but this is not all that occurs. When X
< -e, the graph of f;L2(x) is an increasing function that is concave up if
f)(x) < -1, and concave down if fh(x) > 1. Since fL2(x) has two fixed points,
this corresponds to a two-cycle in f;(x). Figure 427 shows the bifurcation
diagram for the above example, again plotting the fixed point and the periodic
points on the y-axis against the parameter L.
114
,, f() f( )
10.x #-x
Figure 4.26 Graphs of the family f)L(x) =ex for X > -e and X < -e.
-e x---- ýi poi nt
fixed point 1is atractingis repelling
IAwo-cycle
Figure 4.27 Bifurcation diagram for f(x) = Xex with X < 0.
In a period-doubling bifurcation, as the parameter passes through its
critical value, the attracting fixed point becomes repelling and a cyde of period
two emerges. Of course, as we saw with the logistic equation, a bifurcation
can also occur that makes an attracting n-cycle repelling and creates an
attracting 2n-cycle; hence the name "period-doubling." The formal
mathematical characterization of the period-doubling bifurcation is given in
the following result.
115
'THE PERIOD-DOUBLNG BIFURCATION. Given that
1. f;.(0) = 0 for all X in an interval about X0,
2. f,'(0) = -1, and
3. )(f) 2)'/•X I X-)o(0) *0.
Then there is an interval I about 0, and a function h: I-+R, such that
fh(x)(x) * x, but f(x)2 (x) = x.
As shown earlier, without loss of generality we are able to consider functions
with fixed points at the origin and apply the same results in the general case
for arbitrary fixed points. The proofs of both results concerning bifurcations
can be found in Devaney (1989), and involve little more than application of
the Implicit Function Theorem and knowledge of partial derivatives.
A situation where period-doubling bifurcations have been observed
occurs in the field of electrocardiology. Electrochemical events in the heart,
monitored by electrocardiograms (ECG), show periodic activity within the
atria and ventricles of the heart. Abnormal cardiac rhythms, such as
arrhythmia, have long been referred to as "chaotic heart action" (in the
descriptive, non-mathematical sense) by cardiologists. Mostly, these
fibrillations, or chaotic heart rhythms, have been recorded on ECGs
immediately prior to a patient's death. By concentrating on the heart's
natural pacemakers, called ectopic foci and which are located throughout the
heart, cardiologists have found that in arrhythmia, the rhythm of the heart
undergoes a series of period-doubling bifurcations leading eventually to
chaotic behavior (in the precise mathematical sense). While the human
heart requires a much more complex model than the one-dimensional
116
"dynamical system, there is a high correlation between the period-doubling we
have studied and the period-doubling on the ECG. Cardiologists are using
this new theory to try to explain what actually happens in the heart in order
to prevent or cure the condition. A current solution to this problem is to
identify patients who are susceptible to arrhythmia and implant in their
chests a device that detects fibrillations and gives the heart an electronic
"kick" out of its period-doubling path. This device has proven successful, but
it has been difficult to identify the patients who will benefit from its use.
Turbulence is one of the classical and unexplained phenomena in
physics. In the past it has been so inaccessible to physicists and engineers that
systems have been designed normally with "fudge" factors, or factors of safety,
designed to compensate for the effects of turbulence. However, the issue of
turbulence has rarely been directly taken on. Recently, in a physical model
that creates turbulence in a very simple fluid setting, scientists have observed
period-doubling bifurcations in the fluid leading directly to the onset of
turbulence. These observations may be the first steps towards an
understanding of this elusive behavior.
A few more examples of how a changing parameter in the real world can
lead to bifurcations will emphasize the importance of studying this concept.
In the logistic equation it is reasonable to assume that the parameter X, which
measures the species' interaction with its environment, does not change too
drastically from cycle to cycle. On the other hand, in a fluid system, it is just
as reasonable to see how the changing velocity of a flow can change a
parameter quickly enough to induce the period-doubling bifurcations that
lead to turbulence. So it might seem reasonable to surmise that the panel
117
flutter on a supersonic aircraft could be avoided through a careful design that
stays away from dangerous parameter values. However, not every
atmospheric condition can be predicted accurately by a model, or duplicated in
a wind tunnel. Moreover, atmospheric phenomena, such as ice buildup on
wings or wind shears, can quickly push parameters into critical regions. In
complex systems one can rarely anticipate every set of parameter values,
which is an important reason to pay careful attention to parameter space.
J. SARKOVSKIPS THEOREM
We now present another remarkable result about one-dimensional
dynamical systems due to the Russian mathematician, A. N. Sarkovskii. The
development here is from the article by Devaney (1989). First, order the
positive integers in the following manner.
3*5*729s...-
2-3 * 2"5 * 2"7 * 29 '9
22-3 * 225 * 22.7 22-9...9 .
23.3 * 23.5 * 23.7 * 23- *.
.-.. 24 23 * 22 * 2 2 1.
Here, we have first listed the odd numbers in ascending order, followed by
two times the odd numbers, then 22 times the odd numbers and so on,
through every positive integer power of 2 times the odd numbers, finally
followed by the powers of 2 in descending order. Given this ordering of the
positive integers, we present Sarkovskli's theorem.
118
* SARKOVSKII'S THEOREM. Suppose f. R-+R is continuous. If f has a
periodic point of period n, then f also has a periodic point of period k for
all k with n * k in the above ordering.
While this result only holds for one-dimensional systems, it is
remarkably powerful due to its lack of hypothesis. In fact, the only
requirement on f is that it be continuous.
Rather than proving Sarkovskii's theorem itself, we prove a special case
in what follows. This proof is found in the article by Devaney (1989) and
differs from the complete proof of Sarkovskii's theorem by requiring less
bookkeeping. The full proof of Sarkovskii's theorem can be found in
Devaney (1989).
PROPOSITION. Suppose a continuous map f: R-+R has a cyde of period 3.
Then f has periodic points of all periods.
PROOF. The proof depends on two observations. Fu-st, if I and J are dosed
intervals with J I and f(I)nJ, then f has a fixed point on L (Thisis
similar to the result about fixed points we proved in Section HC). The second
observation is as follows: suppose A0, A 1,..., An are dosed intervals such
that f(Ai) Aj+l for i = 0, 1,. .. , n-1. Then there exdsts at least one
subinterval J0 of A0 which is mapped onto A1. Similarly, there is a
subinterval J1 of A1 mapped onto A2, and hence a subinterval J1 of J0 such
that Al m f(J1) and f2(h) = A2. Continuing in this manner, we obtain a
nested sequence of subintervals which map into the various Ai, in order.
119
* To prove the proposition, let a, b, and c be such that f(a) = b, f(b) = c, and
f(c) = a. With only a slight loss of generality, assume a > b > c (the other case
is handled similarly). This hypothesis is shown in Figure 4.28.
t(x),
V so Xc b a
Figure4.28 Athree-cyclewith f(a) =b, f(b)=c,f(c) =a, anda>b>c.
Welet To=[b,a] and Ii=[c,b]. Byour assumptions, f(1O)nII and
f(i) m 16 u Ii. Figure 4.27 shows there is a fixed point between c and b.
Similarly, P has fixed points between a and b, so at least one of these points
must be of period Z Fixing n > 3, we now produce a cycle of period nL
We first find a nested sequence A0, At,..., An-2, of subintervals of I as
follows: let A0 = II. Since f(It) D 11, there is a subinterval Al of A0 such
that f(AO) = A0 =- I. By induction, we can find a subinterval An-2 of An-3
such that f(An-2) = An3, P(An-2) An-4,.. ., and f' 2(An.2) = A0 = 11. Since
f(Ii) D To, there is a subinterval An-i of An-2 such that fn'I(An-1 ) = 10. Finally,
since f(TO) z) II, we have ff(An.1) n II = An-1. Hence, fn has a fixed point in
An-1 from the first observation. We now show this point has period n under
120
f. Since Ij = fi(An-1) for i =0, 1,...,n-2, but I O fn-1(An.-), and fn(An) • I1,
this point has its first n-2 iteratesin Ii,thenjumpsto I0,inthe n-1
iteration, and finally back to I1. This completes the proof.
In addition to the above proposition, another corollary to Sarkovskii's
theorem states: If f has a periodic point which is not a power of 2, then f
has infinitely many periodic points.
Sarkovskii's theorem provides considerable information about a
function. For example, it would be very difficult to check directly whether
the function fRx) = 1 + (5/2)x - (3/2)x2 is chaotic. However, since f(O) = 1,
f(1) = 2, and f(2) -0, the function has a three-cycle. Thus Sarkovskii's
theorem tells us it has cycles of all periods. Hence, we automatically know it
is chaotic without having to check the three defining conditions.
If we consider the period-doubling bifurcations as a route to chaos, then
only finitely many periodic points must have the periods 1, 2, 22, 23, 24,...,
2N for some N. Then as the parameter varies and the dynamics of the
system become more complex, we introduce periods in a specific order 2N+I,
2N+2, .... This argument does not claim that the new orbits appear as period-
doubling bifurcations, but that something similar must occur.
We cannot derive a converse to the theorem from the special ordering. If
we find a cycle of period k, and n * k in the ordering, there is no guarantee
that a cycle of period n exists.
While we used a graphical analysis to show that the Baker map
f(x) = 2x (modl) is chaotic, we now confirm this using Sarkovskii's theorem.
Although the Baker map is not continuous, since every iterate has fixed
121
points on [0, 1], we can apply the above proof of the proposition directly to it.
Thus, if we can find a three-cycle of the Baker map, we will know it is chaotic.
But that is easy: since the points 1/7,2/7, and 4/7 form such a cycle, the
Baker map is chaotic.
Another example using Sarkovskii's theorem to find chaotic behavior is
with the function fRx) = x2 + c, where c = -1.755. Using a computer, we can
verify the attracting orbit of period 3 given by fRO) = -1.755, f(-1.755)
1.325 (to four decimal places), and f(1.325) = 0. This three-cycle guarantees
cycles of all periods and an infinite number of periodic points. Nevertheless,
regardless of the initial value, the orbit is always attracted to this three-cycle.
But then where are the other periodic points if only three are found by
computational iteration? The answer is that all other cycles are repelling and,
because of computer round-off error, iterates always move away from a
repelling cycle (unless the points are rational numbers represented exactly up
to the precision of the computer). This example further illustrates the
sensitive dependence on initial conditions of chaotic dynamical systems.
Now why, of all the infinite orbits, is only one of them attracting? This
question is extremely complicated in general. However, in the case of our
specific example it can be answered using a result from complex analysis. For
a complex analytic map, each attracting orbit attracts at least one critical point
for the map. Now the map f(z) = z2 - 1.755, is an analytic function with only
one critical point z = 0. Hence the function has only one attracting orbit.
122
K THE QUADRATIC FAMILY REVISITED
The last example showed another member of the quadratic family of
maps; namely, f(x) = x2 + c. We now study this map in greater detail as we
vary the parameter c. The results in this section are from Falconer (1990) and
the article by Devaney (1989).
First note that for c > 1/4 the graph of f(x) = x2 + c lies above the
diagonal y-=x and fn(x)-*oo for all x e R. For c = 1/4, the graph of f(x) is
tangent to the diagonal when x = 1/2 (which is a single fixed point). Finally,
for c < 1/4, f has two fixed points which we denote by pi and p2 where
pi <p p. This is an example of the tangent bifurcation as c passes through
the value 1/4. These three cases are shown in Figure 4.29.
f (X) f X) ALf(x)
CA4C -1/4 C < 1/4
Figure 4.29 Graphs of f(x) = x2 + c for c > 1/4, c = 1/4, and c < 1/4.
123
Now observe that for all c<1/4, p2>1/2,so If(p2)9 = 12p21 >1.
However, for -3/4 < c < 1/4, If'(p1) I < 1 since -1/ 2 < Pl < 1/ 2. Finally for
c < -3/4, I f'(pl) I > 1 since pI < -1/2. This demonstrates that after the tangent
bifurcation occurs (as c passes through 1/4) the fixed point Pi is attracting
and p2 is repelling. However, as c passes through -3/4, P1 also becomes
repelling (and in fact, a two-cycle forms around pi). The graphs of fAx) = x2 + c
as c passes through -3/4 are shown in Figure 4.30. As c continues to
decrease, we get another sequence of period-doubling bifurcations, similar to
those we saw with the logistic equation. The frequency of these bifurcations is
also governed by the Feigenbaum constant.
f(x) f(x)
Ptt
C>-314 < -3/4
Figure 4.30 The graph of f(x) = x2 + c as c passes through -3/4.
We have developed three ways to determine if this period-doubling leads
to chaos. The first is to graph f(x), and f2(x) through fn(x), for certain values
of c and observe the fixed points of these graphs. The second way is to solve
algebraically for the roots of Pf(x) = x and observe the fixed points as the roots
124
of these polynomials. For the case f2, solving the equation f2(x) = x yields the
fourth-order equation (x2 + c)2 + c = x, or x4 + 2cx2 - x + c2 + c = 0. Since we
already know two of the roots (pi and p2) for any given value of c, we can
solve directly for the other two. However, as n increases, it becomes very
difficult to solve for the roots of fn(x). Third, we can simply find a three-cycle
for certain values of c (for example c = -1.755) and appeal to Sarkovskii's
theorem as proof that periods of all other orders do exist.
Continuing to analyze the family f(x) = x2 + c, we see from Figure 4.28
that for all c < 1/4, if I xo I > p2, then fn(x0)-+-. Thus, we can focus our
attention on the interval I = [-p2, p2] where all of the interesting dynamics
occur. Let us further restrict our attention to the range of parameter values c
< -2. If we analyze this function on a computer for almost any initial value,
the iterates of f to go to infinity. However, as shown below, there are many
orbits which do not escape under iteration of f.
The graph of f for c < -2 is shown in Figure 431. If we consider the
interval I - [-p2, p2] note that there is a subinterval Ao of I that maps to
values outside of I, hence all points in A0 will escape to Go. Consider the
graph of f2, a similar analysis to the one performed for the logistic map with
X > 4 shows that two subintervals of I - A0 are again mapped outside of I, so
again points escape to @.. The second iteration of f on the interval I is
shown in Figure 4.32. The subintervals that escape I on the second iteration
are labeled A1 and A2 in the figure.
Analyzing the points that remain in I after infinite iteration, we deduce
a Cantor set has been constructed on L While it is not necessarily the classical
Cantor set, nevertheless, it contains no intervals despite an infinite number
125
&f(x) /
(P2, P2)
Figure 4.31 The graph of fRx) = x2 + c for c < -2.
Figure 4.32 The graph of f2(x) for c < -2.
126
'of points. (As we mentioned in Section IIE, this could be an example of a "fat"
Cantor set, depending on the value of c). We now define the set
A={xe I: fn(x)e I V nO}),
and assert that A is a Cantor set.
The dynamics of f on R - A are quite simple because every initial value
tends to +-o under infinite iteration. We want to know what happens to the
set A. To determine this, we simplify the analysis through symbolic
dynamics. Recall that the shift map on code space a. 1-+, is a continuous
mapping. We now try to relate a and f. When we remove the interval A0
from I, two subintervals remain, denoted by I0 and I1 (see Figure 4.33).
Hence, if x e A, then ffn(x) e T0 u 1 1 for all n > 0. Next define the itinerary of
x by S(x)=(sOsls2...) where sie (0, 1) and si = k if and only if fi(x)e Ik.
Sf(x)
(P2' P2)
10 A0 11
Figure 4.33 The subintervals Io and Ii.
127
We now show that S: A-+I is a homeomorphism. First, to see that S is
one-to-one, let x, y e A and suppose S(x) = S(y). Then for each n, fn(x) and
fn(y) both lie on the same side of A0 . It follows that f is monotonic on the
interval between fn(x) and fn(y). Hence, all points in this interval remain in
I0 u I1. This observation contradicts the fact that A is a Cantor set and
contains no intervals, so S is one-to one.
To see that S is onto, for a closed interval J, set fn(j) = (x e I: fn(x) e J). If
I z J, then f'-(J) consists of two subintervals, one in I0 and one in I1. Let
s e I with s = s0s1s2. •., and define
Is...sn = (x e I: x E Iso, f(x) e s1,..., f(x) e Isn,
so Iso...sn = Iso n f-'(Isi) n. . n f-n(Isn). We claim that the Iso...sn form a
nested sequence of nonempty closed intervals as n-+o. Note that
Iso...sn = I6o c f-n(Is5 ...sn). By induction, we assume that Isl...sn is a nonempty
dosed subinterval so that f0 1( _3...n) consists of iwo subintervals, one in To
and one in 11. Hence, Iso...sn is a single dosed subinterval. These intervals
are nested since Iso...sn = I60...sn-1 n fl(Is) c Iso... -1. Hence, the intersection
w.sn is nonempty for any n > 0. Note that if xe r-qQs...n, then
x e Lo, f(x) Ie4,.. ., so that S(x) = sOsls2.... This proves that S is onto.
Since S is one-to-one and onto, it follows that S-1 exists. Hence we need
only show that S is continuous to prove it is a homeomorphism. Thus,
choose xe A andlet S(x) =sOSlSs2.... Let e > 0, and choose n suchthat 1/2n
< e. Consider Ito._tn for all possible combinations of to, t1,..., tn. These sub-
intervals are disjoint, and A is contained in their union. Choose y e A and
8 suchthat Ix-yI <8. Then ye Iso...sn, and S(x) and S(y) agree for the first
n+1 terms. Hence, by the metric on code space -2, d(S(x), S(y)) < 1/2n < e.
128
Thus S is continuous. Trivially, S-1 is also continuous; hence, S is a
homeomorphism.
Since S: A--7 is a homeomorphism , we use topological conjugacy to
show that f has the same dynamics as the shift map a on code space 12. The
commutative diagram for this relationship is shown in Figure 4.34. Since we
know ; is chaotic on L f is chaotic on A through topological conjugacy.
A f -. A
I~s Is
Figure 4.34 The topological conjugacy between f: A--A and a :-4.
We now return to the map f(x) = x2 - 1.755 to learn more about it
through its symbolic dynamics. Since it has a three-cycle, -1.755, 0, and
1.325.. . , Sarkovsldi's theorem guarantees it is chaotic. However, we see we
could have discovered this feature without Sarkovskii's theorem.
We start by finding three open intervals, O1 about 0, 02 about -1.755,
and 03 about 1.325. Select these such that Oi contains the closure of f(OO),
and 01+1 = f(Oi). We can always make this choice because f is a continuous
map (although in practice, the use of a computer would help). Now let I0
denote the dosed interval between 01 and 03 and let Ii be the dosed
interval between 02 and 01. This relationship is shown schematically in
Figure 4.35. We may choose each Qi such that I (f3)'(x) I > 1 on Io u Ii. We
then have f(Io) D II and f(11) D To U I1, so each interval is stretched over its
image.
129
-1.755 0 1.325-. ,( I ) I ) ( I ) .-.
02 1 ] 1 [ 12 3
Figure 4.35 The dosed intervals I and the open intervals 0.
We now introduce symbolic dynamics. Let A = (x: fn(x) e ID u I1 V n 2 0).
We know that A is a Cantor set. To model the dynamics of f on A, consider
modified code space r' where
' = (s1s2s3.. : si e (0, 1) and sk = 0 =* sk+1 = 1),
i.e., this is just 12 with no adjacent pairs of Os. If we now define the map
S: A--+' as above, we see that the condition f(JO) D Ii forces the condition of
no adjacent Os in r'. The diagram in Figure 4.36 shows how f commutes
with a through S. Hence the shift map a: •r'-' provides all of the
information about the dynamics of f. A--A. Thus there exist points of all
periods in r'. In fact, the point 0111.. .10111.. .10..., with blocks of n-I
repeating Is, is the same point found in the proof of the special case of
Sarkovskii's theorem.
A •f .. A
SIs Is
Figure 4.36 Topological conjugacy between f and a.
130
L JULIA SETS
Julia sets, along with the Mandelbrot set, have perhaps been the most
significant factors in generating interest in chaos among laymen. The reason
is because the intricate and beautifully colored computer images shown as"pictures of chaos" are normally pictures of Julia sets. Moreover, the
"movies" of these images, exploding across the screen, are simply the Julia
sets viewed under the continuous changing of a parameter.
Julia sets were actually discovered in the 1920s by the French
mathematicians Gaston Julia and Pierre Fatou. However, their true beauty
and intricate detail were not fully realized until the 1970s when computer
graphics allowed for their inspection in detail. The concept of a Julia set can
be understood with only a basic understanding of complex numbers. On the
other hand, a formal and mathematically rigorous treatment of Julia sets
requires a theory of complex analysis beyond the scope of this thesis. Here we
present only a cursory survey of Julia sets in their ambient space, the complex
plane, still treating one-dimensional maps in the iterated function systems.
While many of the references cited discuss Julia sets, the presentation here is
from Keen (1989) and Falconer (1990). There are many equivalent definitions
of Julia sets, but the one we present is perhaps the simplest to demonstrate
and understand.
DEFINITION. Given a mapping f: C--C of the complex plane, its Julia set
J(f) is the closure of the set of repelling periodic points of f.
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* . As a simple example, consider the map f(z) = 2z. Under infinite iteration
of f, all points in the complex plane excluding the origin tend to o@ (or to be
more precise, the point at o.). The origin is a fixed, and hence periodic point
of this map. Since iteration of any point other than the origin tends away
from it, the origin is repelling. This result is also verified from If'(0) I = 121 >
1. Since the only repelling periodic point of this map is the origin, which is
its own closure, the Julia set for f(z) = 2z is the origin.
We now present a less trivial example which demonstrates many
interesting properties of Julia sets. Consider the map f(z) = z2 + c for c = 0.
All points inside the unit circle I zI < 1 tend to the origin under infinite
iteration. Thus the origin is an attracting fixed point of the map. In fact,
I f'(0) I = 12(0)1 = 0, which also verifies that the origin is attracting. Moreover,
ai points I zI > 1 outside the unit circle tend to -o under iteration of this
map.
Now consider the standard unit circle, I z I - 1. These points are
represented by z - e10 . Then z2 = e020, which is exactly the chaotic map of the
unit circle f: SI-+SI, where f(0) = 20, studied earlier. We know the periodic
points of this map are dense on the unit circle. Since the periodic points of
f(z) = z2 are dense on the unit circle, every point on the unit circle is the limit
of a sequence of periodic points of f. Thus, the closure of the periodic points
of f is the unit circle. Moreover these points are repelling. To see this, recall
that points inside the unit circle converge to the origin, whereas points
outside the unit circle diverge to the point at infinity. Furthermore,
If'(z)l IIzI1. = 12(1)1 =2> 1, verifying that these points are repelling. Hence,
the Julia set for the function f(z) = z2 is the standard unit circle.
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Another concept associated with a Julia set is that of the filled Julia set,
denoted F(f). When the Julia set is a closed curve, the set F(f) is the union of
J(f) with its interior. The filled Julia set is the set of points that do not escape
to - under infinite iteration of f. For the example f(z) = z2, the filled Julia
set is the dosed unit disc, F(f) = (z: Izi < 1). As an aside, the complement of
the Julia set is called the Fatou set and is sometimes denoted F(f) as well,
although Jc is also used. Loosely speaking, J(f) is the set containing the
"bad" (i.e., chaotic) behavior, while the Fatou set is the "good" set, possesing
the well-behaved dynamics.
Having introduced the concepts of Julia sets and filled Julia sets in this
simple setting, we now describe an algorithm for generating computer images
of these objects. If we superimpose the complex axes on a computer screen to
an appropriate scale, then points in the complex plane correspond to pixels on
the screen, although this relationship is certainly not one-to-one. Given a
function f(z), we can iterate each pixel. Since we are interested in the points
that escape to .o, a bound (normally very large) can be set which we call I Z 1,
and above which an iterate is considered as having escaped. Next select k
integers N 1 <N 2 < ... <Nk-=N. Color the screen with k+1 colors based
on the following algorithm: as a point is iterated, if it has not escaped after N
iterations, color it black. If it escapes (goes beyond I Z 1) between 0 and N1
iterations, assign to it another color (say, red). If it escapes between Nj and
N 2 iterations, color it with yet another color (for example, yellow). Continue
in this manner until the entire screen has been colored. Selecting a large
value for k provides more detail, which can be refined further by
experimentally adjusting N and I Z I with respect to each other. The part of
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the screen colored black (if we have chosen N, I Z I, and the scale
appropriately) is the filled Julia set for f, and this region's boundary (provided
it is connected) is the Julia set itself. The colored bands around the Julia set
are contours corresponding to the various escape times of points in the
exterior of the Julia set. The reason the complement of the filled Julia set is
colored is because of the finite scale of the computer screen: there can be great
detail occurring within the area of a single pixel and, while the complete Julia
set is not revealed by just the black area, much can be determined about its
border by examining the distorted contours surrounding it.
For the example f(z) = z2, coloring the screen based on this escape time
algorithm produces a black disc with a sequence of concentric colored circles
around it (which in itself is not particularly interesting). However, recalling
the family of functions fc(z) = z2 + c, as the parameter c is varied some very
interesting results occur. Unfortunately, while this Julia set has many
fascinating properties, an advanced level of complex analysis is required to
establish even its most basic properties. The required concepts include
families of normal functions, the Arzela-Ascoli theorem, and Montel's
theorem. These results are beyond the scope of this thesis, but an excellent
summary of them is found in Falconer (1990), and we provide a synopsis of
them at the end of this section. Nevertheless, we can still provide a brief
description of some of the salient characteristics.
As the parameter c is varied away from the origin, the Julia set (the unit
circle) begins to continuously distort and take on different shapes. Closer
inspection reveals that the boundary appears to become infinitely detailed
and self-similar; in fact, it becomes fractal. (Even with c = 0, the boundary of
134
the filled Julia set is infinitely self-similar, although in a trivial manner.)
While we saw that the dynamics of f on J(f) for c = 0 are chaotic (since they
share the properties of fRO) = 20) they continue to be chaotic on J(f) as c
varies. For the quadratic family f(z) = z2 + c, as c is varied the Julia set varies
from a circle, to a dosed curve with "bulbs" that are "pinched" together at a
single point, to "dendrites" which are fractal structures with no interior, to
"dust" which is a set of disconnected points which are scattered about a region
of the complex plane, similar to the Cantor set. For some Julia sets with
fractal boundaries, like the Koch snowflake, the lengths of the boundaries are
infinite. A further result about Julia sets is that they are either connected
(meaning they consist of one solid piece) or totally disconnected (meaning
they have a structure similar to Cantor dust). The Julia sets for various
values of c are shown in Figure 4.37.
Without going into too much detail, we provide a brief synopsis of the
most important ingredients of the mathematical theory behind these results.
For analytic functions in C (i.e., those that are infinitely differentiable in the
complex sense) techniques of complex variable theory can be used to establish
the basic properties of Julia sets.
It can be shown that an alternative (but equivalent) way of defining the
Julia set J(f) for polynomials f is as the set of all complex values z for which
the family {fk(z)) k = 1, 2,... is not normal. Loosely speaking, a family of
complex analytic functions is said to be normal if it possesses some expecially
strong convergence properties (called "uniform" convergence) on compact
subsets of a given open set. Using a powerful result from complex analysis
known as Montel's theorem, it is possible to demonstrate that if f is a
135
(d)
CeV)
W.4A. . y~ 9,
Figure 4.37 Julia sets of the function f(z) z2 + c for (a) c =-1+ .1i;(b) c =-.5 +.5i; (c) c =-1 +.05i;- (d) c =-.2 +.75i; (e) c =.25 +.52i- (U) c=-.5 + .55i; (g) c = .66i; (h) c = -i. The figure is from Falconer (1990, p. 213).
136
polynomial, then J(f) is non-empty, compact (closed and bounded), contains
no "isolated" points, and much more. Note that this is consistent with what
we have seen with the quadratic family, even in the case when J(f) is "dust."
It should be pointed out that the above results do not necessarily hold for
non-polynomial complex analytic maps. The Julia set for the exponential
map f(z) = ez, for example, is the entire complex plane. Of course,
polynomial functions are not the only ones that generate interesting Julia
sets. Some of the trigonometric families, such as X sin z, also provide very
interesting characteristics as X varies.
M. THE MANDELBROT SET
The Mandelbrot set is often associated with intricate computer graphics.
It has been described from "the most complex object in mathematics" to "the
most beautiful object in mathematics." While Julia sets are found in range
space of a complex function, the Mandelbrot set lies in parameter space,
which is the complex plane when the parameter is a complex number. Like
the Julia set, almost every reference cited discusses the Mandelbrot set.
However, th particular presentation here is based on Branner (1989) and
Falconer (1990). There are two equivalent definitions of the Mandelbrot set,
and both of them are presented here.
One definition of the Mandelbrot set for fc(z) = z2 + c is the set of values
of c for which the associated Julia set J(f) is connected. (This definition
stresses the connection between the Mandelbrot set and Julia sets.) As
mentioned in the last section, the Julia sets for f(z) = z2 + c vary from being
totally connected to "dust." The values of c for which the Julia sets are dust
137
do not belong to the Mandelbrot set. An equivalent definition, which is
perhaps easier to understand, follows.
DEFIMITION. The Mandelbrot set fl is the set of complex values of c for
which the origin does not escape to - under infinite iteration of f(z) = z2 + c.
A picture of the Mandelbrot set in the complex plane is shown in Figure
4.38. The figure is from Falconer (1990, p. 205). We have already seen that for
c > 1/4 on the real line, all values of x including the origin go to infinity
under iteration of fAx) = x2 + c; for c < -2, the origin also escapes. Hence, we
know that the Mandelbrot set contains the interval [-2, 1/4] on the real line.
However, the situation is not nearly this simple when c varies in the
complex plane.
Im
0
! I
-2 -1 0Re
Figure 4.38 The Mandelbrot set.
138
Like Julia sets, the Mandelbrot set is infinitely detailed. In fact, it contains
many smaller copies of itself around its border. However, it is by no means
self-similar because it contains many other interesting shapes. Moreover, its
intricate detail varies significantly among the border regions of the smaller,
Mandelbrot-like sets.
Pictures of the Mandelbrot set can also be generated using the escape time
algorithm used to draw Julia sets. Here the coloring of the complement
becomes particularly important because many of the tendrils extending from
the main body of the Mandelbrot set are too detailed to capture on a computer
screen (regardless of the scale chosen), so they are only evidenced by the
distorted contours surrounding them. It is known that the Mandelbrot set is
connected: even points that appear isolated on computer images are
connected to the main body by dendrites too small to be seen on a computer
screen.
The first definition of the Mandelbrot set shows an intimate connection
with Julia sets of the function f(z) = z2 + c, but the Mandelbrot set contains
even more information about the dynamics of the function f. Once again,
however, any rigorous mathematical development of these dynamics
requires the advanced theory of complex analysis. So again we only describe
some of the more interesting characteristics.
The first result is that each "bulb" of the Mandelbrot set corresponds to an
attracting k-cycle of f(z) for a particular value of k. For example, the large
central cardioid corresponds to the values of c for which f(z) = z2 + c has an
attracting fixed point. To see this, note that an attracting fixed point must
satisfy z2 + c = z and I f'(z) I = 12z I < 1. The boundary of this region is given
139
'by c = z - z2, where I z I - 1/2. In polar representation, this becomes
c = (1/2) e2xiO - (1/4) e4iO 0• 0 < 2x. These values of c trace out a cardioid
in the complex plane with a cusp at z = 1/4 + Oi. Unfortunately, the periods
of the attractive cycles of the other bulbs do not so easily reveal themselves
mathematically.
It is perhaps not surprising that the periods of the bulbs along the real axis
are in direct correspondence with the bifurcations found for the map
f(x) = x2 + c. Recall that for this latter map, a tangent bifurcation occurs at
c = 1/4, and a series of period-doubling bifurcations begins as c decreases
through -3/4. Figure 4.39 shows the Mandelbrot set plotted on the same
coordinate axis as this bifurcation diagram. You can see the alignment of the
main bulbs with the period doubling that occurs along the real axis. The bulb
in the "tail" of the Mandelbrot set corresponds to the three-cycle that emerged
out of chaos around the value c = -1.755 studied earlier.
The Julia sets associated with the c values belonging to the MandelbroL
set vary as the period of the attracting cycle varies among the bulbs. Julia sets
for values of c in some of the different bulbs of the Mandelbrot set are shown
in Figure 4.40. Notice that the number of "bulbs" in the Julia sets that are
pinched together at a single point correspond to the period of the cycles of the
Mandelbrot set. For example, the Julia sets for values of c in the main
cardioid are all simple dosed curves which correspond to the attractive fixed
points, whereas the values of c in bulbs that correspond to attractive n-cycles
have n bulbs converging at a single point. Notice also the very thin Julia set
(dendrite) associated with one of the tendrils of Ml. Dendrites occur for
values of c for which the origin is a periodic point of f(z) = z2 + c; for
140
--0-2 -07
Figure 4.39 The Mandelbrot set plotted against the bifurcation diagram forf(x) = x2 + c. The figure is from the article by Devaney (1989, p. 37)...
example the point c = -i as shown in Figure 4.37. Finally, the Julia set
associated with a point not in the Mandelbrot set is totally disconnected.
Remembering that for values of c < -2, the sets of periodic points for the
iterated maps f(x) = x2 + c were Cantor sets (hence totally disconnected) their
Julia sets, by definition, are also disconnected. Thus these points fail to belong
to the Mandelbrot set.
141
Figure 4.40 The Julia sets of points in different bulbs of the Mandelbrot set(see Figure 437). The figure is from Falconer (1990, p. 214).
Another feature of the Mandelbrot set is the existence of a dense set on its
boundary of points, called Misiurewicz points, for which the image of the
Mandelbrot set in parameter space, and the corresponding Julia set in the
range space, look the same up to a rotation (in a sense that can be made
mathematically precise; see Branner (1989)). Figure 4.41 shows a blowup of
the Mandelbrot set and the Julia set around the Misiurewicz point
c - -.101096 + 1(.956287).
The Mandelbrot set occurs in spaces other than the parameter space we
have presented. In fact, it ajpears to be an almost universal geometric shape.
Recall the coloring of the complex plane through Newton's method for the
function z4 - 1 = 0 in Section IVG. If we color the complex plane for different
142
Figure 4.41 The Julia set and the Mandelbrot set around a Misiurewicz point.The figure is from Branner (1989, p. 103).
values of the parameter X for cubic polynomials of the family P,, chaotic
regions are found between the basins of attraction. However, interspersed
within these regions are small copies of the Mandelbrot set. While these
regions have always existed, it has taken present-day powerful computer
graphics to reveal them. As scientists continue to use computers to examine
dynamical systems more closely, we expect that the Mandelbrot set will
appear with increased frequency.
There are, no doubt, other more fascinating properties of the Mandelbrot
set yet to be discovered or proven. Each property reveals something about the
complexity of the iterated map f(z) = z2 + c. While we have not discussed all
known results here, this cursory summary does provide considerable insight
into the complexity of this chaotic mapping.
While the function f(z) = z2 + c appears to be a very specific form of the
quadratic family, it is in fact topologically conjugate to every quadratic
function for various values of c. To see this, consider the function
H(z) = az + P with cc# 0. Then h1l(f(h(z)) = (d2z2 + 2al3z + p2 + c -J)/c.
Appropriate choices for the values a, P, and c produce any quadratic
143
'function whatsoever. Thus, in studying the dynamics of f(z) = z2 + c, reveals
the entire family of quadratic functions.
N. THE SMALE HORSESHOE
Our attention so far has been restricted to one-dimensional dynamical
systems. In so doing, we have learned much about chaos. However, since
many real-world phenomena occur in two and three dimensions, the range
of applications has been restricted. We now investigate our first two-
dimensional dynamical system, the Smale horseshoe. Instead of developing
the horseshoe algebraically, a strict geometric interpretation of the map is
given. The primary references for this section are Holmes (1989),
Guckenheimer (1990), and Devaney (1989).
The Smale horseshoe was originally constructed to help interpret the
periodically forced oscillator, which commonly appears in applications in
physics, mechanics, and electrical engineering. Normally, the systems under
investigation are modeled with ordinary differential equations, and the
Smale horseshoe turns out to provide an intuitive way to see why the
equations sometimes lead to chaotic behavior.
Many versions of the Smale horseshoe exist. We present here the
version that is the simplest geometrically. Thus, take the unit square in
Figure 4.42, stretch it out by a factor of three in one direction, and
simultaneously shrink it by a factor of three in the other direction to obtain a
long bar. Then bend the middle section of the bar into a horseshoe and
superimpose it back on the original square, as shown in the figure. Denote
this geometric mapping by F. Notice that the two shaded bands do not escape
144
'the unit square under this first iteration. Their preimages are the horizontal
bands shown in the figure.
Because the preimage of F can be determined precisely, it is an invertible
map. Thus it is possible to study not only the forward orbit of points, but
their backward orbits as well. We are interested in finding the invariant set of
the unit square under the forward and backward orbits of F. These are the
points which do not escape the unit square under infinite forward and
backward iteration. Then we will be able to investigate the dynamics of the
particular physical system associated with the Smale horseshoe by studying
the dynamics on this invariant set.
D STEP2 STRETCH A
A B STP3A B
STEP 1START
C D C D
C D B
Figure 4.42 Construction of the Smale horseshoe.
145
Now, iterate the map a second time, as shown in Figure 4.43. Observe
that the image of the shaded area of the first iteration, and its preimage appear
as before.
PREIMAGE
C G H D
CG HD BF EA
Figure 4.43 The second iteration of the Smale horseshoe.
By superimposing the image of F on its preimage for the first two iterates, we
construct geometrically an invariant set, shown as the darkly shaded region
in Figure 4.44. Here we label the horizontal and vertical bands H and V,
respectively.
146
... ..... 1H(1)
HI1
H(1 0)
H(0) H(01)
H(00)
V(0) V(1) V(00) V(01) V(10) V(11)
Figure 4.44 The invariant set of the Smale horseshoe.
Notice that the set of points A remaining in the unit square under
infinite forward and backward iteration has a Cantor-like appearance. In fact,
that set turns out to be the direct product of two Cantor middle-thirds sets.
The variations of the Smale horseshoe mentioned earlier involve using
different values for shrinking and stretching the unit square under F, as well
as using a different placement of the horseshoe when it is superimposed back
on the square. All variations, however, still create Cantor-like invariant sets.
In order to understand the dynamics of this system, we only need to
analyze the dynamics on the invariant set (since all other points escape under
iteration for F). To undergo this analysis, first note that the forward and
backward orbits of any point x in the invariant set also belong to it.
Specifically, each point in these orbits is in one of the horizontal bands HM or
Hi. Hence, define the mapping S: A--Z by the rule Sj(x) = i if Fj(x) e Hi for
i e (0, 1). Thus, every point x in the invariant set is associated with an
infinite string of indices of the horizontal bands to which it is mapped under
F. Noticethat the index of points in E runs j=...,-3,-2,-1,0, 1,2,3, .... So
unlike code space Z2 for one-dimensional maps (which consisted of semi-
147
'infinite sequences), the space I consists of bi-infinite sequences. Figure 4.45
shows a point x and its orbit under three forward iterates and one backward
iterate. So for j -1..,-, 0, 1, 2,3,..., Sj(x) =... 00100... from the
horizontal bands in which each iterate lies. Noticing that Sj(F(x)) = Sj+l()
one sees that F applied to the set A corresponds to the shift map a on the
space Y, Moreover, every symbol in 1: corresponds to a unique orbit of F,
because every image V completely intersects its preimage RL Therefore, the
mapping F and the shift map a on infinite code space are topologically
conjugate through the map S, as shown in Figure 4.46.
The horseshoe map has been very useful in analyzing physical systems
because it extends to any Euclidean space Rn. The connection with ordinary
differential equations is through a concept known as the Poincare map. If the
phase space associated with an ordinary differential equation is intersected
with a plane normal to any orbit, then the orbit intersects the plane exactly
once during each cycle. The collection of these points of intersection is called
the Poincare map. While the horseshoe map was constructed originally in
connection with the Poincare map of a periodically forced oscillator, there is a
general method for finding horseshoes that applies to a wide range of
Poincare maps. The procedure has helped scientists and engineers
understand the dynamics of the associated physical systems.
The actions of stretching and bending in the Smale horseshoe are
frequently encountered in physical systems. Predicting the orbit of points in
such systems (a simple taffy pull serves as a classical example) has always
proven elusive. The science of chaos has helped explain why these systems
have been so difficult to understand.
148
H W
". .Fl ... ...x )E
FHlxH0 •~f •S
F Mx Fz :(x)
Figure 4.45 The orbit of a point x of the invariant set.
A F- AILs Is
Figure 4.46 The topological conjugacy between F: A-+A and a:. 7-1.
O. THE HIENON MAP
With the Smale horseshoe providing a geometric introduction to two-
dimensional dynamical systems, we now turn our attention to another map
of the plane that exhibits many of the interesting properties of two-
dimensional maps. The material in this section is presented as a series of
exercises in Devaney (1989), to which most of the answers and results come
from Rasband (1990), Alligood (1989), Cherbit (1991) and Moon (1987).
149
"The Henon map Hab: R2-+R 2, is defined by the equations
xi =1 + yo- axo2,
yi = bxO.
Notice that H depends on two parameters, a and b, and that it has only one
nonlinear term (x2). Thus H is one of the simplest higher-dimensional
nonlinear maps we can study. A number of questions regarding the Henon
map have not been resolved because of the wide range of possible parameter
values, but for certain parameter values it exhibits some very interesting
behavior.
First, note that the Henon map can be expressed as the composition of
three maps H 3oHoHi, where Hi(x, y) = (x, 1 - ax2 + y) is a nonlinear bending
(and a quick check with calculus shows it is area preserving); H2(x, y) = (bx, y)
is an expansion or contraction in the x direction, depending on the value of
b; and H3(x, y) = (y, x) flips the contracted, bent image about the main
diagonal.
The case b = 0 makes the Henon map topologically conjugate to the map
g(x) = 1 - ax2 if we consider the projection of H onto the x-axis. For the case
lb I > 1, the map H 2 is not a contraction and the iterates diverge. Hence, we
restrict our attention to the range 0 < I bI < 1.
Now fix b. It is easy to show the fixed points of the Henon map are
(x, y) = [b- 1 ± ([b- 112 + 4a)1 /21/2a, bx).
A doser inspection reveals that for (b - 1)2 + 4a < 0, or a < -(b - 1)2/4, these
points have an imaginary component yielding no fixed points in R2.
Moreover, when a = -(b - 1)2/4, the fixed points coincide (so there is only one
attracting fixed point). Finally, for a > -(b - 1)2/4, there are two distinct fixed
150
"points, one of which is attracting. Here, for a fixed value of b, as the
parameter a increases through a critical parameter value, a tangent
bifurcation occurs.
As the parameter a continues to increase a series of period-doubling
bifurcations appears eventually leading to chaos. Let a. denote the value of
a beyond which chaos occurs. Then the dynamics of the Henon map can be
determined geometrically in a familiar setting. For a fixed value of b, let R
be the larger root of a42 - (b - 1)4 - 1 = 0. Let S be the square centered at the
origin with vertices at (±Wk, ±R). Figure 4.47 shows the images of S under H
for a < a.. and a > a.. Note also the effects of H1, H2, and H3 in the way the
square S is bent, contracted, and flipped. Additionally, for a > a., the
geometric construction looks similar to the Smale horseshoe (and, in fact, it is
a horseshoe). Thus the dynamics of the map H for a > a. (for a fixed b) are
indeed chaotic.
'•- • 1. X - X
a<a a>a
Figure 4.47 The images of S under the Henon map.
151
The particular value b = 0.3 has been studied extensively and the tangent
bifurcation occurs at a = .1225. If a increases holding b = 0.3 constant, in the
range 1.052 < a : 1.082 a series of period-doubling bifurcations occurs which
eventually lead to chaotic behavior.
A particularly interesting phenomenon occurs close to b = 0.3 and a = 1.4.
Here we have an attractor of the system. The infinite iteration of bending,
shrinking, and flipping the plane yields results not yet fully understood.
Nevertheless, with the aid of computers, it has been possible to compute
these results numerically and view them graphically. Iterating an initial
point (xN, yo) under H yields a set of points, called the attractor of H, that
appear to be invariant under infinite iteration of R The attractor of H for
the values b = 3 and a = 1.4 is shown in Figure 4.48. The dynamics on this
attractor are chaotic (as just shown geometrically with the analog to the Smale
horseshoe). Numerically it has been found that the attractor appears to have
a dense orbit, sensitive dependence on initial conditions, and to be
topologically transitive. However, since the evidence of this invariant set has
only been suggested by the use of numerical computation (and not established
with any mathematical rigor) many of its properties are still not dearly
identified.
The Henon attractor (if it truly exists) for b = 3 and a = 1.4 fits into a
class of attractors referred to as strange attractors. While a formal definition
of strange attractor has notbeen developed to date, there are three conditions
that seem to be characteristic of them. These characteristics are:
i. Points "nearby" the attractor converge to the attractor under infinite
iteration of the function.
152
2. The dynamics of points on the attractor are chaotic.
3. The attractor has a non-integer fractal dimension.
S..-: I..°°
S-• /•- -
Figure 4.48 The Henon attractor. The figure is from Holden (1986, p. 90).
By "nearby" we refer to a region of the plane called the basin of attraction,
inside of which all points converge to the attractor. The basin of attraction
depends on the particular function, but in the case of the Henon attractor it
turns out to be the entire Euclidean plane.
Magnification of the Henon attractor indicates that it is infinitely detailed,as evidenced by the "bands" in Figure 4.49 actually being composed of smallerbands of points. Additionally, its fractal dimension has been estimated
numerically at 1.26 for the parameter values b = .3 and a = 1.4.Nevertheless, considerable mystery remains concerning the Henon attractor
(as well as many of the other interesting strange attractors that have been
153
discovered numerically or physically). Because of their structure and self-
similarity, fractal geometry is currently being applied to the study of strange
attractors.
L6, b-
L3 LM
U.'
Lt
L17
:0 C9il .S L-N ILV l11U r.ll
Figure 4.49 Magnification of the Henon attractor. The figure is fromBerge (1984, p. 133).
While strange attractors come up in models of physical equations such as
the Duffing equation, the van der Pol equations, or the Rossler equations,
they have also been'seen in physical systems. While many infectious diseases
appear to follow definite cycles, measles appears to follow a strange attractor
with fractal dimension 2.5 when viewed in the proper phase space.
Additionally, Saturn's rings, because of their remarkable resemblance to the
strange attractors of many mathematical systems, are being studied in this
new light in great detail (however, this connection is still being investigated,
and no conclusions have yet been drawn).
A final remark about the Henon map: if we set the parameter b = 1, the
map becomes an area preserving map of the plane. Since the map Hl(x~y)=
154
'(x, 1-ax2 + y) is area preserving as observed earlier, we see that with b = 1,
H 2(x, y) = (x, y) and H3(x, y) = (y, x) also preserve areas. This condition leads
to an entirely new set of phenomena, one of which we briefly mention here.
As the parameter a increases, orbits of different periods are created, but the
last orbit to develop is a two-cycle. This provides an example of where
Sarkovskii's theorem fails to apply in two dimensions.
P. THE LORENZ EQUATIONS
It is appropriate to conclude our mathematical treatment of chaos with
the Lorenz equations because they comprise one of the first systems to bring
chaotic dynamical systems to the attention of the mathematical community.
The primary references for this section are Sparrow (1982), Holden (1986), and
Fischer (1985), although some of the presentation follows that of Berge (1984),
Guckenheimer (1990), and Thompson (1989).
The Lorenz equations have been studied extensively since the mid 1970s,
and numerous interesting results have been derived from them. However,
to discuss many of these results requires mathematics beyond the level of this
thesis. We present here a cursory summary of some of the results which are
consistent with the mathematical level of this thesis, and particularly those
which relate to some of the material we have already discussed. A rigorous
mathematical derivation of the results we present here can be found in
Sparrow (1982).
The Lorenz equations were developed in an attempt to model the earth's
atmosphere to simulate weather patterns using a small computer. The
Lorenz system is defined as follows:
155
dx/dt = -ox + ay
dy/dt = rx - y -xz
dz/dt = xy - bz,
with a, r, and b positive parameters. This is an example of a continuous
dynamical system. This system models a flat fluid layer being heated from
below and cooled from above (representing the Earth's atmosphere being
heated from the ground's absorption of sunlight and losing heat into space).
In the resultant temperature flow, x represents the convective motion, y
represents the horizontal temperature variation, and z represents the
vertical temperature variation. The parameters u, r, and b are related to the
Prandtl number, the Rayleigh number, and the size of the region being
modeled (see Figure 4.50).
COOL PE BOUNDARYY
WARM LOWER BOUNDARY
Figure 4.49 The model for the Lorenz equations.
The Lorenz system is a very crude model of weather dynamics and is of
little practical value. Actually it has been studied most extensively for
parameter values that are nowhere near those of the Earth's atmosphere.
While the system does have physical relevance to the Maxwell-Bloch
equations for lasers, and to convection problems in specially shaped regions
156
(usually toroidal), they attract the most attention because of the wealth of
information they provide about dynamical systems.
For one-dimensional or planar systems of differential equations, the
Poincare-Bendixson theorem (see Hirsch, 1974, p. 248) guarantees that one can
completely classify the solution, as to whether it approaches a fixed point or
limit cycle, or goes to infinity in a finite amount of time. However, there is
no analogous theorem in three dimensions where many systems with
interesting behavior are being discovered. The Lorenz system is of great
mathematical interest because it possesses many of the characteristics of other
higher-dimensional systems. This is not to say it is typical, as it has some
distinct characteristics (for example, symmetry) but it does demonstrate
characteristics typical of many general higher-dimensional systems.
Because the original paper on this subject by Lorenz (1963) fixed the
parameter values at r, = 10 and b = 8/3 and investigated the system as the
parameter r varied, much of the literature has taken this same approach, as
we do here. Hence, we consider the system
dx/dt = 10(y- x),
dy/dt = rx - y- xz,
dz/dt = xy -(8/3)z.
First note the apparent simplicity of the system. There are only two
nonlinear terms, xz and xy. Also, there is a natural symmetry to the
equations given by (x, y, z,) -+ (-x, -y, z). The z-axis is invariant because
points which start on it stay there and tend towards the origin. Moreover,
when x = 0 the dx/dt term carries the same sign as y so that all points
157
which rotate about the z-axis do so in a clockwise manner when viewed from
above z = 0.
The model has no solutions which tend to infinity; in other words, there
is a surface inside of which all solutions tend towards the origin and herein
all solutions remain. To see this, consider the ellipsoid f(x, y, z) = x2 /2o +
y2/2 + z2/2 - (r + 1)z - p = 0 for p arbitrarily large. We show that the dot
poduct of the velocity vector and the outward normal vector to the ellipsoid
is always negative, i.e., Df = (dx/dt)fx + (dy/dt)fy + (dz/dt)fz < 0. Substituting
into the Lorenz equations we obtain Df = a(y - x)fx + (rx - y - xz)fy + (xy - bz)fz.
Then substitution of the ellipsoid partial derivatives yields If = -x2 - y2- bz2 +
(r + 1)bz. For large enough I in the equation for the ellipsoid, the quadratic
term in z in the expression for If always dominates the linear term in z, so
for this surface the flow is always towards the origin. Hence, no trajectory
originating a finite distance from the origin will go to infinity.
We now analyze the system for a =10 and b = 8/3 as we vary r. To
begin, we restrict our attention to small values of r (i.e., r < 30). A quick
check shows the origin is a fixed point for all parameter values, but we would
like to know whether it is attracting or repelling.
We introduce here the concepts of stable and unstable manifolds. A
stable manifold of a point p is the set of all points that tend to p in forward
time (as t-). The unstable manifold of p is the set of points that tend to p
in backward time (as t--+--). For our purposes, a manifold can be thought of
as simply a surface in phase space. These manifolds can be determined from
the eigenvalues of the linearized system near the point p. The linearized
system has the matrix:
158
'°0
r-z- -x,
y x -b
which, when evaluated at the origin yields
The eigenvalues of this system are X1, X2 = (112)[-o-1± ((a - 1)2 + 4or)1/21, and
X3 =-b.
Since, for r < 1, we have [(a - 1)2 + 4cr'l/2 < [(- 1)2 + 40]1/ 2 = a + 1, so all
three eigenvalues of the linearized system evaluated at the origin are
negative. Hence, the origin is globally attracting. The phase portrait of this
condition is shown in Figure 4.51. However, for r = 1, the eigenvalues
evaluated at the origin are X1 = 0, X2 = -a - 1, and . 3 = -b. This zero
eigenvalue is analogous to the nonhyperbolic fixed point we encountered in
our study of discrete systems. We require more theory to determine whether
the manifold associated with this eigenvalue is stable or unstable. For r > 1,
the eigenvalues are X1 > 0, X2 < 0, and X3 < 0. Since two of these eigenvalues
are negative and one is positive, the origin has a two-dimensional stable
manifold and a one-dimensional unstable manifold for r > 1.
As r passes through 1, not only does the origin become unstable, but two
new fixed points are introduced at (±b(r - 1)1/2, ±b(r - 1)1/2, r- 1) which we
denote C+ and C-. You should recognize this as a bifurcation. A similar check
as we did above of the linearized system near these points shows that they
159
have complex eigenvalues. Furthermore, for values of r < rH (as defined
below), the real parts are all negative. Hence, these points are attracting. The
phase portrait of the system for 1 < r < rH is shown in Figure 4.52.
0
Figure 4.51 The origin is globally attracting for r < 1.
C" 0 c+
Figure 4.52 Phase portrait of the Lorenz system for 1 < r < rH.
Numerical solutions to the Lorenz equations indicate that for
1 < r < 13.926, orbits on the unstable manifold of the origin tend directly to the
nearest attracting fixed point C+ or C-, as indicated in Figure 4.53. However,
for r > 13.962, these orbits "cross over" and are attracted to the other stable
point (see Figure 4.54).
160
z+
Unstable manifoldof the origin
/S table manifoldof the origin
Figure 4.53 Solution trajectories for r < 13.962.
Unstable manifoldA " of the origin
table manifoldof the origin
Figure 4.54 Solution trajectories for r > 13.962.
161
Since we know the stable manifold of the origin is planar near the origin
and includes the entire z-axis, and since trajectories cannot cross each other,
the stable manifold must be twisted in some strange way. What happens here
is the stable and unstable manifolds of the origin merge and form an orbit
called a homoclinic orbit. A homodinic orbit of a point p is a set of points
that tend to p in both forward and backward time (see Figure 4.55). The
introduction of a homoclinic orbit is another example of a bifurcation.
Figure 4.55 The homoclinic orbit of the Lorenz equations.
As r continues to increase, we note that at r = rH = [G(a + b + 3)]/(o - b -1),
the real parts of the complex eigenvalues of the linearized system at C+ and
C- cross the imaginary axis and become positive. This is another example of
a bifurcation as C+ and C- become unstable. Hence all three fixed points are
now repelling. For o= 10 and b = 8/3, rH - 24.74, numerical solutions
indicate that for r > rH, there is an attractor (called the invariant set) to which
all solutions tend as t-+-.
162
Figure 4.56 shows this invariant set for r = 28, b =8/3, and a = 10 as it is
projected onto the xz-plane. This invariant set is a strange attractor and it
exhibits some interesting properties. For example, the trajectory continues
forever within the bounds shown, yet never crosses itself or returns to the
same point in space. Additionally, the dynamics on the attractor are believed
to be chaotic, although for continuous systems more theory is required than
developed in this thesis.
40 130o
z
20
10
-10 0 10 20X
Figure 4.56 The Lorenz attractor. The figure is from Holden (1986, p. 126).
There is no closed form solution to the Lorenz equations, so most of the
evidence as to their behavior has been obtained numerically and is
163
conjectured. However, the following results for small values of r have been
verified computationally numerous times, and are widely accepted as the
system's true behavior:
1. For r < 1, all solutions tend towards the origin.
2. For 1 < r < 13.926, all trajectories spiral into one of the attracting fixed
points C+ and C-.
3. For 13.962 < r < 24.06 an invariant set appears in the trajectory, and some
solutions wander among the invariant set before spiraling into either C+
or C-. The closer r gets to 24.06, the longer some solutions stay near the
invariant set.
4. For r > 24.06, some trajectories stay forever near the invariant set,
although for r < 24.74, some trajectories eventually spiral into C+ or C-.
For r > 24.74, the fixed points C+ and C- become repelling, and all
trajectories remain forever near the invariant set. These invariant sets
are similar to the one shown in Figure 4.56, and get closer to it as r
increases.
Using an analysis similar to the Poincare map, we can see a further
connection between continuous and discrete systems. Considering the
homoclinic orbit of the Lorenz equations, we can construct a small box about
the origin, and analyze where orbits near the homoclinic orbit penetrate this
surface. This analysis shows variations on many of the exotic structures we
studied for discrete systems, including horseshoes and Cantor "books" or
"fans," which are families of two-dimensional Cantor sets "sewn" together
164
along a one-dimensional manifold, or "spine." These intriguing results are
all presented in Sparrow (1982).
For large values of r, numerical solutions to the Lorenz equations have
been obtained that exhibit many of the phenomena we studied earlier for
discrete systems. For values of r in certain intervals (called "windows"),
stable periodic orbits develop that bifurcate as r decreases through the
window. One such window appears at 99.542 < r < 100.795. For
99.98 < r < 100.795, the orbit shown in Figure 4.57 appears. In the interval
99.62 < r < 99.98, a different orbit appears which has two "loops" that pass very
close to each other (see Figure 4.57). This is a period-doubling bifurcation as r
decreases through the critical value r = 99.98. An entire sequence of period-
doubling bifurcations occur as r decreases from 100.795 to 99.542. If we let rn
be the values at which these bifurcations occur, then evaluation of the ratio
(rn-1 - rn)/(rn - rn+j) yields approximately 4.67 in the limit, which is very dose
to the Feigenbaum constant.
PAR 140-
*1- j12.300 7 100-
7 so-
X 80
-2o 0 20 -23 0 20
Figure 4.57 Orbits for r = 100.5 and r = 99.65. The figure is from Sparrow(1982).
165
A second window appears from 145 < r < 166. The numerical solutions
for r = 160 and r = 147.5 are shown in Figure 4.58. Again, period-doubling is
apparent. A final window occurs for 214.364 < r < ,. Period-doubling is
again observed in the solutions for r = 350, 260, 222, and 216.2, as shown in
Figure 4.59. Additionally, a symmetric orbit is seen for r = 350.
200-175
I
150
I00 - , * ' ' ' " 0 - "" " .10. . .
S40 -20 0 20 40 -40 0 0 00
Figure 4.58 Orbits for r = 160 and r ,147.5, showing period-doubling.The figure is from Sparrow (1982).
.30
Figure 4.59 Orbits for r :350, 260, 222, and 216. The figures are fromSparrow (1982).
These are just a few examples of the many observed phenomena of the
Lorenz equations. Additionally, because of the wide range of parameter
values, there are even more unanswered questions about the system.
However, in light of what we studied for discrete dynamical systems, these
results have a direct analogy to the discrete phenomena we studied earlier.
Although the Lorenz equations reveal very little about the weather, they
do give considerable information concerning continuous dynamical systems,
a small amount of which was discussed here.
The behavior of the system does tell us that weather is unpredictable to
any degree of accuracy projected for any large amount of time into the future.
From what we know about chaotic dynamical systems (and weather is surely
chaotic), even if we were to develop an accurate model and measure
atmospheric conditions accurately on an arbitrarily small grid, the sensitive
dependence on initial conditions of the system causes any computed
(predicted) solution to stray arbitrarily from the actual weather, given the
slightest reading error. Additionally, small perturbations (which could never
be modeled) such as a single person lighting a match, could cause the whole
system to follow a new orbit. On the other hand, if weather follows some
strange attractor on which the dynamics of the system are chaotic, then not
only is it unpredictable (sensitive dependence on initial conditions), but every
type of weather possible (topological transitivity) could be experienced.
Moreover, there will be a dense period, providing some order to the weather
allowing us to predict such things as seasonal changes.
167
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169
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170
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